predictive models for drillin-g thrust and torque - a comparison of three flank
TRANSCRIPT
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8/11/2019 Predictive Models for Drillin-g Thrust and Torque - A Comparison of Three Flank
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Predictive Models for Drilling Thrust and Torque - a comparison of three Flank
Configurations
E. J. A. Armarego
(1).
University of Melbourne/Australia; J. D. Wright, Engineer, Government Aircraft Factory,
Melbourne/Australia
Pred ic t i v e model s fo r d r i l l i ng t h rus t and to rque a r e p re sen t ed and compared fo r t h ree d r i l l f l ank
The models a re based on the mechanics of cu t t i ng a na ly si s , fundamenta l machining data
The th ree f l an k shapes i nves t i ga t ed a r e
The e f f ec t s o f f eed , speed and prominen t d r i l l po i n t geome t ri ca l f ea tu re s on t he p red i c t ed t h r us t
The di fferences in
coni ca l f l ank ' and the oth er two f la nk models has
also
been shown to be small
c o n f i g u r a t i o n s .
such a s
t h e s h e a r s t r e s s a nd c hi p l e n g t h r a t i o a s w e l l a s t h e f l a n k c o n f i g u r a t i o n w hi ch a f f e c t s t h e
bas i c t oo l ang l e s l i k e normal r:ke a t t he ch i s e l edge reg ion .
the 'Plane Flan k ' , the popular Conica l Flank , and the C lea rance p l anes f l ank
.
and to rque re su l t ed i n p )aus ib l e and comparab le t r ends fo r t he t h ree f l a nk shapes .
predic t ions be tween the
fo r t he work ma te r i a l t e s t ed . In add i t i o n , good cor re l a t i o n between p red i c t ed and expe r imen ta l l y meas-
ured th ru st s and torques has been found fo r a wide range of cond i t io ns.
I t i s shown th a t p rov ided the ba s i c geomet ry a t t he d r i l l cu t t i n g edges can be e s t ima ted , t he more
complex d r i l l f l ank ana lyse s a re no t e ss en t i a l fo r adequa t e t h ru s t and to rque p red i c t i ons when us ing
t h e d r i l l i n g a n a l y s i s p r e s e nt e d .
INTRODUCTION
The th r us t , t o rque and power i n d r i l l i n g are
impor tan t mach in ing pe r fo rmance ch a ra c t e r i s t i c s
re-
qu i red f o r improvement s i n mach ine t oo l and d r i l l
designs as
w e l l
as f o r t he se l ec t i on o f op timum cu t t -
i n g c o n d i t i o n s .
With the expected larg e r i ses i n t h e
p r o p o r ti o n o f t h e t o t a l a v a i l a b l e p r o d uc t io n t i m e
s p e n t i n
metal
rem oval when u s in g CNCJDNC mac hin e
tooLs
l ]
t h e r e i s cons ide rab l e scope fo r r educ t ions
i n m ac hi ni ng c o s t s t o o f f s e t t h e h i gh c a p i t a l i n v e s t -
ment of modern manu fac turi ng syst ems. Thus the need
fo r machining performance da ta has become in crea sin gly
more important i n manufactur ing.
The development of methods fo r o bt ain in g machin-
ing performance data
i s
complic ated by th e numerous
var ia ble s to be considered. These inc l ude the many
t o o l a nd c u t g e o m e tr i c a l v a r i a b l e s , t h e r e s u l t a n t
c u t t i n g v e l o c i t y a nd t h e msterial p r o p e r t i e s o f t h e
to ol and workpiece . The popular empir ica l approach
o f o b t a i n i n g t h e r e q u i r e d d a t a d i r e c t l y f ro m e x p e r i-
ments usual ly allows f o r
a
few of th e more obvious
va r i ab l e s ( e .g . f e ed , speed) fo r e ach work
material
and machining opera t ion tested.
r e s u l t s f o r d r i l l i n g a nd o t h e r o p e r a t i o n s h av e b e en
compi led ove r yea r s o f t e s t i ng L2 ,3] , t he se r iou s
s h o r t a g e o f d a t a
i s
e vi de nt i n t h e l i t e r a t u r e .
Based on a ser ies o f i n v e s t i g a t i o n s , a mechanics
o f c u t t i ng approach fo r p red i c t i ng t he fo rce s and
power i n d r i l l in g and oth er common opera t io ns has been
proposed
[ 4 ] .
This approach involves the development
o f o r thogonal and ob l ique cu t t i n g ana lyse s fo r t he
machining ope ra t io n consider ed. The fo rce and power
equa t ions de r ived inc lude t he t oo l and cu t geome t r i ca l
v a r i a b l e s , t h e r e s u l t a n t c u t t i n g s pe e d a nd b a s i c p ar a -
meters
of t h e cu t t i n g mode ls . For p red i c t i on purposes
t h e b a s i c c u t t i n g pa ra m et er v a l u e s , s u ch a s t h e s h e a r
s t r e s s I
a nd c h i p l e n g t h r a t i o r , ar e found from
' c l a s s i c a l ' o r th o go n al c u t t i n g e i p e ri m e n t s w h i l e t h e
norma l r ake ang le and ang le o f i n c l i n a t i o n are obtained
from an ana ly sis of the to ol geometry and
i t s
s t a n d a r d
s p e c i f i c a t i o n .
The u s e of t h i s a p pr o ac h f o r d r i l l i n g f o r c e s
i s
dependent on a knowledge o f t he d r i l l p o in t geome try
and i t s spe c i f i c a t i on . Although the r e i s cons ide rab l e
agreement about th e genera l appearance and spe ci f ica t -
i on o f t he d r i l l po in t [2-91 t he p rec i se geome try i s
unknown [9 ,10 ]. The geometry
a t
t h e d r i l l l i p s p r e-
sen t s no d i f f i c u l t i e s , howeve r, t h e f l ank geome try
w hi ch a f f e c t s t h e c h i s e l e d ge r e g io n i s e s s e n t i a l l y
unsp eci f ied and dependent on th e sharpening method
used.
The
s i m p l e s t
f l a n k s h a p e , c o n s i s t i n g of
a
s in g l e p l ane , ha s been used wi th cons ide rab l e success
f o r f o r c e p r e d i c t i o n s
[ll].
From s tud i e s o f po in t
sharpening methods th is 'Plane Flank ' shape has been
shown t o b e u n ac c ep t ab l e f o r g e n e r a l pu r po s e d r i l l
produ ct ion whereas th e popular con ica l g r indin g method
seems most s ui ta bl e and i s commonly used i n pr ac ti ce
[ l o ] . Never the l e ss , o the r g r ind ing methods and f l ank
s h ap e s a r e a l s o u se d i n p r a c t i c e [ 9 , 10 ], w h i l e
a t t empt s t o develop f l ank shapes f o r improved d r i l l -
l i f e h a v e b e en r e p o r t e d 1 1 2 3 . I t
i s
t he re fo re impor t -
a n t t o s t u d y t h e e f f e c t s o f d i f f e r e n t f l a n k s h ap e s o n
the t h rus t and to rque.
I n t h i s p a pe r p r e d i c t i v e mo de ls f o r d r i l l i n g
th ru st and to rque based on th e above approach are pre -
sen t ed and compared fo r t h r ee f l an k conf igura t i on s .
These
shapes i nc lude t he y rev ious ly r epor t ed 'P l ane
f l a n k '
Ell],
t he popu lar
Although many us ef ul
Conical Flank' and a
Clea rance P l anes F l ank ' .
THRUST
AND
TORQUE
ANALYSES
The t h r u s t a nd t o r q u e a n a l y s es f o r t h e t h r e e d r i l l
f lan k shapes can be developed by consid er ing t he two
d i s t i n c t r e g i o ns of t h e d r i l l , na me ly ; t h e l i p r e g-
ion and the ch i se l edge reg ion .
f i g u r a t i o n doe s n o t a f f e c t t h e l i p r e g io n t h e a n a l y s i s
a t t h e l i ps w i l l be common fo r a l l t he d r i l l f l a nks
cons ide red .
On
t h e o t h e r h an d s e p a r a t e a n a l y se s f o r
the ch i se l edge reg ion w i l l be necessa ry fo r e ach
f l a n k s ha p e t o c a t e r f o r d i f f e r e n t c h i s e l e dg e
geometry.
t o t h a t r e p o rt e d e a r l i e r [4.115.
Thus as shown in
Fig.
1 ,
the deformat ion process
a t
t h e l i p s
i s
t r e a t e d
as
a number o f c l a s s i c a l o b l ique cu t t i n g e l ement s ,
each wi th d i f f e re n t normal r ake ang le y , i n c l i n a t i o n
ang le
x
and re su l t an t cu t t in g ve1ocity"V depending
on th e hean rad ius a t e ach e l ement .
The
glementa l
deformation fo rce components AF AF and AF as w e l l
as th e elemental edge for ce com%&enes a r e cgnverted
in to the elementa l th ru st ATh and torque
AT
. By
summing th e e lemen ta l forc es &he to ta l thrus h
Th
and
to rque
T
a t t h e l i p s a r e es t ab l is h e d. I n th e l f p
reg ion t i e s t a t i c o r ' t o o l ' g eo me tr y pr o vi d es a s u f f -
i c i en t ly accura t e r ep re sen t a t i on o f t he dynamic o r
'working' geometry.
Since the f lank con-
The gen era l an al yt ic a l a proach used i s s i m i l a r
The ch is e l edge region i s al so analysed by consid-
e r i n g
a
number of elements, however the deformation
process may be cons ide red a s c l a s s i ca l o r thogonal cu t t -
ing wi th hig hly ne gat ive normal rake a ngle and low
r e s u l t a n t c u t t i n g s pe ed ( i . e . t h e i n c l i n a t i o n a ng l e x s
can be shown to be ze ro o r very
small
f o r t h e f l a n k s
s tu d i ed ) . In t h i s r eg ion the dynamic ang le s canno t be
ignored . Fur the r t he e l ementa l fo rce s
are
found from
empi r i ca l fo rce /wid th da t a ob t a ined f rom nega t ive r ake
or thogona l cu t t i ng t e s t s where d iscon t inuous ch ip
format ion occurs.
c o n f i g u r a t i o n
i s
fou nd by summing th e common th r u s t
a nd t o r q u e a t t h e l i p s t o t h e c h i s e l e dg e t h r u s t a nd
t o r q u e o f t h e r e l e v a n t d r i l l f l a n k s h ap e .
The Lip Region
The s a l i e n t g e o m e t r ic a l f e a t u r e s , e l e m en t a l f o r c e
components and ve lo ci t y vecto r f or a s e le c te d e lement
are shown in Fig. 1 . For s impl i c i t y t he fu l l de fo rm-
a t i on geome try and a ssoc i a t e d cu t t i n g ana lys i s vec to r s
( e . g . s h e a r f o r c e , s h e a r an d c h i p v e l o c i t i e s ) a t ea ch
e l ement a re omi t t ed .
The t o t a l t h r u s t a n d t o r q u e f o r e a c h d r i l l f l a n k
These de t ai ls may be found i n
[ill.
For any e lement , say the
j t h
element f rom the
ou te r cor ner , th e e lementa l thr us t ATh and torqu e AT
due to th e deformat ion proces s and the 'edge for ces ark
given by:
AThtj
=
2[(AF +AF )c os cs in p - (PFR+BFRE) c o ~ A ~ C O S ~
Q QE
+ s inX s in p s in c )
AT
=
2r(AFp
+
aFpE)
.ej
where the symbols re le vant to th e e lement consi dered
a re de sc r ibed in t he nomencla tu re .
l i ps a re equa l and expre ssed by
The wid th o f cu t Ab fo r a l l t he ML e l em e nt s a t t h e
Ab
=
[Dcosw~-D'cosu']cosAs/(~M~sinp)
3)
where w
=
sin-l(ZW/D) 4 )
(5)
6)
* ' =
7
- = 1
D
=
Lc
=
2W/sin(n-p)
=
2WJsinly'
The radius
r
a t
t he mid-po int o f t he cu t t i ng edge
o f
t h e j t h e le m en t i s t h e r e f o r e :
r
= t[Dcoswo/2 - ( j -%)AbI2 + W2)
( 7 )
Annals
of
the CIRP Vol. 33/1/1984
5
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Tr ue v i ew
of
VW and
A2
A S
FIG 1. Sal i ent Feat ures
for j tn El ement at
t he Li p
Sect i on XX
AFp, AFpE
The cor r espondi ng equati ons f or t he geometr i cal quant -
i t i es of t he j t h el ement r equi r ed befor e usi ng equat-
i ons (1) and (2) can be shown to be
w = s in - l (Wr)
(8)
5 =
t an- ' [ t anwcosp]
10)
(12)
6 = t an- 1[2rt an60/ D] (9)
asD
= s si n- l [si np si nw] (11)
yref
= tan-1[tan6cosw/(sinp-cosp si nw t an61
'nD
t
=
f s i np cosc/ 2
=
'n = ' ref -
A L = Ab/ cosi (15)
AA = t Ab (16)
Fr omt he si ngl e edge obl i que cut t i ng anal ysi s [ 4. 13]
appl i ed to each el ement t he def or mat i on f or ce compon-
ent s and basi c cutt i ng par ameter s ar e r el at ed by t he
expressi ons
A F ~ TAA[ COS A~- Y~) COSA~+ t anncsi nhssi nl n] / B (17)
AFQ
=
TAA si n(An- yn) / B
(18)
A F ~ T AA[ c o s ( A~- Y ~) s ~~X~t annccoshscosXn]/ B (19)
tan+ =
r ~ ( c o s ~ C / c o s ~ s ) c o s ~ n / [ l - r ~ ( c o s n c / c o s ~ s ) s i ~
ta
=
t anAcosnc (20)
tan(++An)
=
tanascosyn/(tannc-sinyntanis) (22)
( 21
r'
wher e
B =
[ cos2( $n+hn
-yn) + tan2ncsin2hn]4sin ncoshs
(23)
The el emental edge f orce components ar e f ound f r om
AFpE = KIP Ab
AFQE =
KIQ
Ab
AFm
and
=
KI R Ab
*
o
The t hrust Th and t orque T
f or t he whol e l i p r egi on
i s f ound fromthe summati on' of al l t he el ement s, 1. e.
Combi ni ng t he above equat i ons suggest s t hat t he t otal
l i p thrust ThI Land t orque TIL may be expressed as
Th, and
T,
= f uncti ons (D, 2W, 2p ,
d o ,
4 f ,
M,
'I,
r I L ,
.
K l p8
K l Q )
(29)
For quant i t ati ve f orce predi cti ons al l the above var -
i abl es must be known or f ound. The fi r st si x quant i t -
i es r epresent t he known speci f i ed dri l l poi nt f eat ur es
and t he f eed f , whi l e M i s sel ected to adequatel y
al l ow f or t he vari at i on4 i n cut t i ng geomet ry w t h rad-
i us al ong the l i ps. The r emai ni ng f i ve basi c quant i t -
i es are f ound f r omt he cl assi cal ort hogonal data at t he
appropr i ate condi t i ons f or each el ement deri ved f r om
t he f i r st seven var i abl es i n equat i on (29). For any
gi ven work mater i al , t he basi c cut t i ng quant i t i es,
e. g. r , wi l l depend on t he nor mal r ake angl e y =
a r esuf t ant cutt i ng speed V
.
For exampl e, wheR
ynD
cut t i ng
1020
St eel [ l l ] t heef ol l ow ng equati ons have
been f ound f r ommul t i - var i abl e r egr essi on anal ysi s of
ort hogonal cut t i ng t est data:
r =
0. 3427
. 00292
y
+ . 00315 V (30)
T
= 5 1 2 . 9 ~1 0 ~1. 319
xnD106y ( N/ m2y (31)
= 32. 84 + , 559 Y - (de@e) (32)
= 8 4
280
-
1, 397 (N;) . (33j
El p
=
631100
-
716 ynD (N/ m ( 3 4 )
1Q
The Chi sel Edge Regi on f or t he Pl ane Fl ank Dr i l l
consi st s of a st r ai ght chi sel edge perpendi cul ar t o
t he dri l l axi s bounded by the t wo pl ane f l anks as
shown i n Fi g.
2.
The st ati c normal r ake angl e
y
i s
negat i ve, const ant and numeri cal l y equal to hal f nt he
wedge angl e y at the chi sel edge for a l l radi i .
Si m l arl y thewst ati c nor mal cl ear ance angl e a
const ant and equal to t he compl ement of y
r adi i .
f or al l radi i as evi dent i n F i g. 2. Froma geometr i c
anal ysi s
of
t he Pl ane Fl ank dr i l l t he wedge angl e 2y
i s r el ated t o the speci f i ed poi nt angl e 2p and chi sey
edge angl e onl y [10,11] s o t hat when t he r esul t ant
cutt i ng speed angl e n i s al l owed f or, t he dynam c
angl es y and a can be f ound and wi l l var y w t h t he
radi us w%le &hen8ynam c i ncl i nati on angl e
s D
i s
const ant at 0
.
The r el evant equat i ons ar e
For t he Pl ane Fl ank dr i l l t he chi sel edge regi on
i s
f o p al l
The stat i c angl e of i ncl i nat i on
w
i s zero
YnD
= n
-
Yw
(35)
anD = 90
- yw
- n
(36)
yw = tan- ' [ tanpsi n(n-$)] = tan- ' [t anpsi n$' ]( 37)
n - tan- l [Vf/ Vw] = tan- ' [f / 2nr] (38)
wher e
Fi g. 2. Chi sel Edge Geomet r y - Pl ane Fl ank Dri l l
( i . e. i n the regi on r O
d8kat i on process when u 1 . e . Ocr zr . However
si nce r i s a ver y smal l "1roport i on of t ge cki sel edge
l engt h L[ll] t he l att er pr ocess can be negl ected.
For any kt h el ement f r omt he chi sel edge corner
as shown i n Fi g.
2,
t he el ement al chi sel edge t hr ust
AThPt and tor que T^^ are expr essed as
where AF and AF ar e t he ei emental f orce components
consi st i & of t heQEombi ned def or mat i on and edge f or ce
component s. The wi dt h of cut Ab f or al l t he M el em
ent s wher e or t hogonal cut t i ng occur s ar e equa 'and
gi ven by
wher e r t he r adi us r and cut thi ckness t at t he
m dpoi nk' of el ement k are f ound f r om
Ab
=
[Lc
-
2rL] / 2Mc = [D
-
2rL] / 2Mc
41 )
r L = f t anp si n$' / Zn (42)
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r L / 2
-
(k - &)&b
t
-
fCcosrl/2
(43)
(44)
From orthogonal cut t ing considera t ions
A F ~ ~[FpC/b]Ab = ClpAb (45)
A F ~ ~[ F / b ] ~ b
=
CIQPb (46)
Q
The to t a l ch i se l edge t h rus t Th
the o r thogona l cu t t i ng reg ion i g t h e r e f o r e :
k=Mc k=Mc
and torque T c due to
ThC k AThck ; Tc =
kil
ATck
4 7 ) I(48)
Combining the above equat ions the c hi se l edge thr us t
and torque fo r the Plane Flank d r i l l become
=
ThC and
Tc
functions (2W, 2p,
w
f ,
M c , Clp .
1Q
(49)
For quan t i t a t i v e p red i c t i on purposes t he t h r ee spec i f -
i ed d r i l l po in t f e a tu re s 2W, 2p and
,
the feed f and
M are known or se l ec te d whi le
C
and C are found
f fom cu t t i ng da t a a t t he cond i t i hks r e l e@nt t o each
elemen t. For 1020 s t e e l [ll] C and C
are
r e l a t e d
to t he cu t s i z e and rake ang le &ord in iQto t he
equat ions
:
ClP - [FpC/b] = 1 .188x106t'651(90+)0 D) ' 06
CIQ = [FQC/b]
=
19.14x106t '635(90+f' D )- '6 20
(50)
(51)
The Chisel Edge Region for the Clearance Planes Flank
The t h r u s t a nd t o rq u e a n a l y s is f o r t h i s d r i l l
f l ank conf igura t i on
i s
e s s e n t i a l l y s imi lar t o t h a t f o r
t h e P l an e F la nk d r i l l e x ce p t t h a t t h e r e l e v a n t a n g l es
e . g . y y should be used. The s a li en t geometry i s
shown f n r i g . 3 . The chi se l edge region i s r ep re sen t -
e d by a s t r a i g h t c h i s e l ed ge p e r pe n d i cu l a r t o t h e d r i l l
axi s bounded by two planes i n the v ic in i t y of the
ch i s e l edge . The f l ank ad j acen t t o each l i p is a l s o a
p l ane i nc l i ned by t he l i p c l ea rance angle
CL
a t t he
out er corner . The corresponding planes a t t8e
l i p
and
c h i s e l ed ge f o r e a ch d r i l l f l a n k i n t e r s e c t a t a l i n e
through the ch is e l edge corner and perpendicular to
t h e c h i s e l e dg e as shown i n Fig. 3 . Hence the l i p
c l ea rance ang le CL and the ch i s e l edge s t a t i c norma l
c l ea rance ang le an a t t he c h i se l edge co rne r a re
equa l .
ang le
a
as w e l l as the wed
e
angle"2y
are
cons t an t
for a1l"points on the chi sef edge .
Bovh s ta t i c and
dynamic inc l in a t i on angles and X D , r e s p e c t i v e l y ,
a r e Oo so t h a t t h e c u t t i n g a g t i o n i 8 orthogonal wi th
highly negat ive rake angles when
a
>O. From the
g e o me t ri c al a n a l y s i s o f t h e d r i l l @ Tn t t h e l i p c l e a r -
ance angle
C L
a t a ny r a d i u s r on t he l i p
i s
given by
tanC . = cotpsinw+coswsecwo[tanCeo-(2W/D)cotp] (52)
A t t he c h i se l edge co rne r w=w'=n-) a nd t h e l i p c l e a r -
ance angle C a equals an hence from equation (52)
tanan = cotpsiny-cos dsecwo [tanCao- (2WID) cot p]
and from Fig. 3 the wedge ang le 2yw and st a t i c normal
rake
y,
are given by
The st a t i c normal rake angle
a
and c learance
(53)
2yw
=
n - 2an (54)
Y
= -rw (55)
The l im i t i n g rad ius
rL
when anD = 0 i s expre ssed a s
(56)
coso
rL = ~n~cotpsin coswo-cos~~tanCeo-2Wcotp/D]
The ch is e l edge th rus t and torque can be predic ted
from the previous equat io ns (35) t o
(51)
provided yw
and
r
from equat ions (53) 5 4 ) and (56) are used
ins tek d of those f rom equat ions (37) and (42) .
genera l expression fo r ThC and
Tc
comparable to
equa t ion (49) a r e
Thc and Tc
=
func t ions
(2W.2p,~,CLo,f,Mc,Clp,ClQ)
57)
By adding the t ot a l c hi se l edge thr us t and torque to
t h e c o r r es p o nd i n g v a lu e s a t t h e l i p s t h e t h r u s t a nd
t o r q u e f o r
the
Clea rance P lanes F l ank d r i l l a s a whole
i s ob ta ined .
The Chisel Edge Region for the Conical Flank D r i l l
The ch i s e l edge t h r us t and to rque ana lys i s fo r
t h e c on i ca l f l a nk d r i l l i s a l s o
s im i l a r
t o t h a t f o r
t h e
Plane
F l an k d r i l l a l th o u gh
the
geometr ica l anal -
ys is fo r the c learan ce , rake and wedge angles a t each
element i s more in t r i c a t e t han fo r t he P l ane Fl ank
d r i l l . A s shown i n Fig .
4
and d iscussed i n [9.10]
the ch i s e l edge on ly approx ima te s a s t r a i gh t l i ne . The
i nc l i na t i on ang le s and
a
a re ve ry
small
f o r a l l
points on the chise lsedge sb th at the proc ess may be
cons ide red t o be o rthogona l cu t t i n g i n t he r eg ion
a
20. Therefore the s t a t i c normal rake and normal
c%.rance an gl es may be approximated by th e re le va nt
ang le s i n t he sec t i o n ing p l anes such a s AA i n F i g .
4 .
S in ce t h e c o n i ca l f l a n k s i n t h e v i c i n i t y o f t h e c h i s e l
edge may ac t as
f ace s o r f l anks , t he ang le be tween the
t angen t t o t h e cone 2 f la nk and the plane normal t o
The
SECTION
AA
SECTION B B
FIG.3. Chisel Edge Geometry -
Clearance Planes Flank Vw
SECTION
AA
Cone 2 Apex
2.
FIG .4. Ch is el Ed e Geometry
-
Cone 1
Apex
t h e d r i l l a x i s i n F i g. 4 r ep re s en t s a whi l e t he
cor re spond ing ang le wi th r e spe c t t o c8ne 1 i s the
complement of I y 1 ( i .e . 9O-ly I ) . Both a and a
w i l l vary alo ng Phe ch is el edgg as do y, a8d
Y
FBI.
C o n i c a l d a n k
.I
From the con ica l gr indin g analy sis [9] th e equat -
i o n s
of
t he ch i se l edge and the two d r i l l l i p s
are
given by
sinecose 2 sinecose 1,
Y[-CxCy(sinxcosx)2 '(cxcy)
(
~ , 2 ( c o s 2 e - cos2x)cos2e
(59)
(60)
x = -W/cosXg - y ta n a f o r l i p
1
x = ~ / c o s h - ytanXg for l i p 2
where
B , x . X
C and
C
are the conica l gr inding method
parameters .g'ByXnumerieally sol ving equat ion (58) and
(59) or (60) tho ch is e l edge corner radi us rc- L c / 2
and i t s x,y co -ordina tes can be found.
c l ea rance ang le
a n
on the fl an k produced by cone
2 i s
given by:
tanan =
- C O S Y "
tank z+tan$ (cos2
x
- s in2
x
tan28 )
- tanv2 tany" sinxcosxsec2e]
+ [ a m 2 ( s i n 2
- cos2xtan2e)+sin~cosxsec2e
(61)
g g
A t
a rad ius
r
a s i n F i g .
4
t h e s t a t i c n o rm al
where
tan$" = x/y (62)
tanhz
=
(63)
and sin%osXsec20 - Jsec*x2(cos2Xsec2e-l)+sin2xsec2e
tanvz
=
By symmetry the ang le correspon ding t o
a
(61) fo r th e cone 1 fla nk rep re se nt s thencomplement
of
I y
1
i . e .
90
-
l y n ( .
cX + (x2+y2)'sin*"
c - (x2+yz)%osy"
cX +
r sin*"
C~ -
r
cosq'l
-
Y
(64)
-
cos2x sec2e
i n e q u at i o n
Hence by replacing ( 9 0 - ( y
1
an8 u 1 f o r a n , a z and V P i n equa t ions (61) and ( 4 )
7
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where i s given by
t anh l =
Cx -
r
cosv"
I.
r
s in , "
Y
e x p r e s s io n s f o r t h e c a l c u l a t i o n o f i ar e found. The
dynamic normal rake and clearance angles, and the
wedge angle
2vw
a r e
Y n D
= n - i Y n / (66)
anD
=
a n - (67)
2vw
=
n / 2
-
u n j Y n /
(68)
Because of th e complexity of t he above equ atio ns
t h e l i m i t i n g r a d i u s r
,
where a
i s
ze ro i n equa t ion
( 6 7 ) i . e .
a
equat io n. fn st ead , numerica l methods must be used
invo lv ing equa t ions (58) (61) t o (64) t o ob t a in
r
and
a a s
w e l l
as equat ion (38) for
I
i n e a ch i n t e r a t i o n .
FP%m equat i on (58) t o (67) t he above an aly si s toget her
wi th equa t ions (38) t o (41) , (43) t o
(51)
from the
Plane F l ank ana lys i s t he e l emental and to t a l ch i s e l
edge thru st and torqu e can be evaluate d.
I t i s
i n t e r -
e s t i n g t o n o t e t h a t f o r t h e c o n i c a l f l a n k t h e e q u a ti o n
comparable to equat i on (49) i s
Thc and T c = func t ions ( 2 W , 2 p , ~ , C r , , D , 6 , f , M ~ ~ ~ , C ~ ~ )
Fo r p r e d i c t i o n p ur p os es t h e d r i l l s p e c i f i c a t i o n f e a t -
ures 2W,2p,w,Ce
,
D as w e l l a s t h e c o n i c a l g r i n d i n g
parameter
e
muse be known.
rep or t s [9 .10] t he usua l method o f spec i f y ing t he d r i l l
p o i n t i n v ol v in g t h e f i r s t f i v e v a r i a b l e s i n e q u at i on
(69) a re no t s u f f i c i en t t o un ique ly de t ermine t he con-
i c a l g r i n d i n g p a r am e te r s s o th a t one of th ese parameters
0
must be p re se l ec t ed . The feed f , M a n d b a s i c c u t t -
i ng da t a C and C need to be known'as fo r the oth er
f lan k shap& consih?red.
Again by summing the c hi se l edge and l i p t hr us ts
and to rque t he va lues fo r t he d r i l l a s a who le can be
ob ta ined .
COMPARISON OF MODELS AND PREDICTIONS
=
q ,
cankot be exp@ssed by a si ng le
(69)
A s no ted i n p rev ious
A
qu a l i t a t i ve compari son of t he t h re e f l ank con-
f ig ura t i on s ha s been made f rom cons ide r a t i ons o f t he
above ana lyse s and the p red i c t ed t h ru s t and to rque
t ren ds. For qu an t i t a t i ve comparisons the magni tude
of t he p red i c t ed and expe r imen tal t h r us t and to rque
have been obta ined for a 1020 s t ee l workp iece wi th t h e
cha rac t e r i s t i c s given in equa t ions (30) t o (34) and
(50) (51) .
Due to the complexity o f t h e d r i l l i n g p r o ce s s
geome try t he t h re e mode ls ne ces s i t a t e compute r a ss i s t -
ance fo r t h ru s t and to rque p red i c t i on s . I t
i s
a l s o
e v i d e n t t h a t a l l th e models all ow f or th e many geomet-
r i c a l d r i l l p o i nt f e a t u r e s , t h e f ee d a nd t h e c u t t i n g
speed V (when the ba s i c c u t t i ng pa rame ter s i n equa t -
ion (307 ar e use d). Combining equ ati ons (29) and (49)
the t o t a l t h r us t and to rque fo r t he P l ane Fl ank mode l
can be expressed by:
Th, and
T t =
func t ions
( D ,
2W, 2p , 6 0 , c , f , MQ, M
T I
r Q ,
Kip.
KlO.
C l p . Clo)
P i O )
Thus fo r t he P l ane F lank d r i l l
da
i s n o t ' i n c l u d e d i n
th e model. For th i s d r i l l flan k Zhape the common
d r i l l s p e c i f i c a t i o n g i ve n by D , 2
w,
2p,
6
4 and C r
i nc ludes a r edundan t f ea tu re ( i . e . t h e d r i ' i i shape
i s o
ove r - spec i f i ed ) and
C P
i s dependent on the oth er
fe at ur es 2p, 2W, D andow [ 1 0, 1 1] . F u r t h e r t h i s d r i l l
shape has unacceptably high C t val ues (230 to 35O)
when a l l th e ot he r fea tu re s l ie 'within the recommended
v a l u e s f o r g e n e r a l p u rp o se d r i l l s [ l o ] .
The co r re sponding to t a l t h rus t and to rque func t -
i o n s f o r t h e ' C l e ar a nc e P la n es F l an k ' d r i l l
i s :
Tht and T t
=
funct ions (D, 2W, 2p , 6
T hi s d r i l l f l a n k sh ap e al lo w s f o r a l l t h e s i x s p e c if i ed
d r i l l p o i n t f e a t u r e s so tha t the geometry i s uniquely
d e s cr i b ed i n t h e v i c i n i t y o f t h e l i p s a nd c h i s e l e d g e.
Th i s approx ima te r ep re sen t a t i on o f t he d r i l l geome try
can apply for a v ar ie t y of f lan k shapes away from the
c u t t i n g e d g e w i t h o u t a f f e c t i n g t h e
f o r c e p r e d i c t i o n s .
F or t h e c o n i c al f l a n k d r i l l t h e t o t a l t h r u s t a nd
torque functions become
:
Th, and T t
=
fu nc ti on s (D, 2W. 2p, S o $,Ce , e
Although a l l t h e s i x p r om in en t d r i l l p o i n t f e a t u r e s
are
i n c l u de d t h e s e do n o t u n iq u el y d e s c r i b e t h e d r i l l
po in t geomet ry gene ra t ed by t he con ica l g r ind ing
method 191. Thus one of t he grin din g paramet ers such
as the semicone angle a must be known o r s el ec te d and
c ou ld a f f e c t t h e t h r u s t a nd t o r q ue as no ted i n equa t -
i on (72) . This model i s a u s e f u l r e p r e s e n t a t i o n
o f
many popu lar d r i l l po in t g r inde r s when the se a re s e t
a c c o r di n t o t h e s t r a i g h t l i p de s ig n co n ce p ts r e p o r te d
i n C9.147.
d r i l l and cu t geometry have been compared f or th e
w C ro , f ,
M I , Mc, , r r , A , Kip,
tiQ,
l p , C I Q (71)
f , M a . Mc, T , r rBA,KIP,
K I P ,
Cyp.blQ) (73
The th ru s t and to rque t r ends wi th va r i a t i on s i n
t h r e e f la n k m od el s. U si ng a t y p i c a l d r i l l s p e c i f i c a t -
io n (D 12. 78 m W / D
=
.14, 2p
=
120, p
=
1300,
1 = 300 and C r = 120) each var ia ble was independent-
18
al te re d over 'a wide range (e .g . 2p from
1100
t o
1400,
6
f rom 200 to 350) and th e th ru st and torq ue
when d r i l l i n g 1020 s t e e l we re no t ed .
The th re e models
gave very sim i lar t ren ds f or a l l the common va r i abl es
i n t h e a n a l y s e s . F or a l l t h e f l a n k s h ap e s t h e t o t a l
t h r us t Th inc rea sed wi th i nc rea se s i n
D . 2W/D.
and
(except f& r the Plane Flank where
'r'
had l i t t l e e f f e ct )
and dec rea se s i n he l i x ang le 6
.
For t he c l ea ran ce
p lanes f l ank and con ica l f l ank ' d r i l l s , ve ry sma l l i n -
c r e a s e s i n t h r u s t o c c ur r ed a s t h e l i p c l e a r a n c e a ng l e
Cr
dec rea sed whi l e i nc rea se s i n
e
(35O -
500)
only
maggina lly i nc rea sed t he t h ru s t (
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._
- - -
30
r
2or
0
f
=
5 . 4
i
m d
I I
h
10
1 4 2 - 6 - 2 2 6
10
1 4 1 8
Plane Flank-Conica l Flank Clearance Planes Flank -
7. obs
Conical Flank
Je
= 1 . 4
J E
0 . 7
30
r
10
2ol
0
- 1 0 1 2 3 %
- 1 0 1
2 3
FIG.5. Comparison of Pre dic tio ns based on Conica l
t o t he con i ca l f l ank a r e a low
3 . 3
and
5.4 .
r e s n e c t -
Flank Model.
i v e l y .
15%. The torque comparisons show even smaller
d i f f e r e n c e s w i t h
E
of 1.4 and .7 f o r t h e P l a n e
Flank and Clearance Planes Flank d r i l l s comparisons
r e s p e c t i v e l y , w i t h a l l - E v a l u e s
less
than 3 . The
s l i g h t l y h i g h e r E and E v a l u e s f o r t h e t h r u s t t h an
the t o rque a r e r easonab le s ince t he ch i se l edge and
i t s
geometry contr ibutes
a
h ighe r p ropor t i on o f t he
t o t a l t h r u s t t h an t h e t o t a l t o r q u e . N e v e r t he l e s s ,
the th re e f lank models give comparable predic t ions
when app l i ed t o gene ra l pu rpose d r i l l s .
and experimenta l t hr us t and torqu e has a ls o been made
us ing t he c on ica l f l ank model as r e f e r e n c e - E i g h t
d r i l l s w i t h a w id e ra ng e o f d r i l l p o i n t
f e a t u r e s
i l : 117.7 ' - 1 3 7 . 4 0 ;
C
: 12.6' - 20;9O, 6
.
1 0 . 7 - 3 0 . 8 9
have been se l e c t ed f ro& ba t ches o f as pr8duced man-
u f a ct u r e d d r i l l s a n d t e s t e d o n 1 2 s t e e l a t t h r e e
feeds
( .
1 0 2 , . 2 0 4 and . 3 0 6 mm/rev) and one speed of
1 8 . 3 m/min. For eac h t e s t c o n d i t i o n f i v e c u t s w e re
t aken to improve t he e s t ima te
of
t he measured t h rus t
and to rque . The pe rcen t age d i f f e rence i n p re d i c t i on
i . e . ( (Predic ted-Experimenta l )
x
lOO/Experimental) and
i t s average value f or th ru st and torque have been used
to a sse ss t he mode l. Very rea sonab le co r re l a t i o n
between predic ted and experiment va lues
have been
found. For t he d r i l l on
a
whole , t he ave rage pe rcen t -
a ge d i f f e r e n c e i n p r e d i c t i o n
w a s
- 7 .9 f o r t h e t h r u s t
and
- 2 . 2 %
fo r t he t o rque us ing a
s e m i
cone angle e
=
3
The
i nd iv idu a l pe rcen t age d i f f e r ence
w a s
w i t h i n
20% f o r t h e m a j o ri t y o f t h e t h r u s t a nd a l l b u t two o f
t h e t e s t c o n d i t i o n s f o r t h e t o r q ue . T he se r e s u l t s
compare favourably wi th simi lar comparisons wi th a
d i f f e r e n t s e t o f d r i l l s [ll]. Further informat ion on
t h es e r e s u l t s i s g i ve n i n [ 1 5 ] b u t th e f u l l d e t a i l s
w i l l
b e p u b l is h e d i n a l a t e r p a p e r . I t
i s
i n t e r e s t i n g
t o n o t e t h a t t h e d r i l l f l a n k s ha pe s o f m os t o f t h e
' a s p roduced ' gene ra l pu rpose d r i l l s t e s t ed canno t be
gua ran t eed t o be con ica l f l ank d r i l l s . Neve r the l e ss
th e measured ch is e l edge wedge ang les
2 y
a t t he d r i l l
dead cen t re were wi thin
60
of t he p red i c red va lues
used i n t he c oni ca l f lank model which ranged from
104.6O t o
112.3O.
ment be tween t he th ree f lan k models as shown i n
equa t ions (73) t o (78) and th e h i s togram in F ig .
5,
i t
i s
e v i d en t t h a t t h e o t h e r
two
f la nk models
w i l l
y i e l d s i m i l a r l y a c c ep t a b le c o r r e l a t i o n w i th t h e
expe r imen tal t h r us t and to rque a s no t ed fo r t he con-
i c a l f lan k model . Thus for f orce (and hence power)
pr ed ic t io ns , any of t he th ree models may be used.
However, t he ch oic e of the most appr op ria te model i s
open
t o
some deb ate. The Pla ne Flank model i s t h e
s imple s t o f t he t h re e bu t exc ludes
Ca.
. The conica l
flank model
i s
th e most complex model'considered and
th e semi-cone ang le e should be known. For tuna te ly ,
any rea sonab le e s t ima te of e w i l l s u f f i c e f o r f o r c e
p r e d i c t i o n s a l th o u gh
e
may become importan t f o r ot he r
performance measures s uch
as
d r i l l - l i f e . T h e Clear-
ance Planes Flank model i s appea l ing due t o t h e r e l a t -
iv e ly simple geometry which in c ludes a l l the 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 . The p r e f e r r e d mo del
may be th e one which adequat e ly descr ibes the c hi se l
e d ge ge om et ry a nd i n c o r p o r a t e s 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 o f r e l ev a n ce t o t h e f o r c e s i n d r i l l i n g .
The 'Clearance Planes Flank' model seems the most
s u i t a b l e
of
t h e t h r e e mo de ls s t u d i e d p a r t i c u l a r l y
s i n c e t h e a c t u a l d r i l l f l a n k g eo me tr y o f m a n uf ac tu r ed
d r i l l s i s no t p rec i se ly known o r s t anda rd i sed .
conica l f lank model i s unnecessar i ly complex for
The percentage di fference E a r e a l l l e s s khan
A
qu an t i t a t i ve comparison between the pred ic ted
(D:6.35 - 1 2 . 7 IIUII; 2W/D:
.12
-
, 2 2 8 ;
2p: 112O - 120.5',
I n v i ew o f t h e q u a l i t a t i v e a n d q u a n t i t a t i v e a g re e -
The
force p red i c t i ons bu t shou ld be pe r s i s t ed wi th fo r
d r i l l l i f e s t ud i e s due t o t h e p o pu l ar i ty o f t h i s d r i l l
poin t gr ind ing method.
CONCLUSIONS
based on mechanics of cu tt in g analy ses and fundamental
cut t in g data have been developed, compared and te ste d
f o r t h r e e d r i l l f l a n k c o n f i g u ra t i on s .
t he
l i p
c l ea rance ang le C r
,
th e Clearance Planes
Flank' model includes a l l ehe s i x s p e c i fi e d d r i l l
poi nt fe a tu re s whi le th e 'Conica l Flank ' model requires
the semi-cone angle
e
t o b e known i n a d d i t i o n t o a l l
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 . N e v er t he l es s t h e
th ree mode ls r e su l t ed i n comparabl e t h rus t and to rque
pre dic t ions when numerica l ly t es ted over
a
wide range
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 u r t h e r , good c o r r e l -
a t i on between predic ted and experimenta l da ta has
been obta ined.
type t h ru s t and to rque equa t ions i ncorpor a t i ng t he
many d r i l l and cut ge ometr ica l va r ia bl es have been
e s t a b l i s h e d
f o r
u s e i n i n d u s t r y .
I t i s shown tha t provided th e ba si c geometry a t
t he d r i l l c u t t i ng edges can be adequa t e ly mode ll ed,
the mechanics of cu t t i ng approach can be succ essf ul ly
used to p r ed i c t t he t h rus t and to rque wi thou t r e so r t -
i ng t o t he more complex d r i l l f l ank geome tr i ca l
ana lyse s .
Acknowledgement. The fi na nc ia l sup por t rec eiv ed from
t h e
Australian Research Grants Scheme
i s
g r e a t l y
a p p r e c i a t e d .
Pred i c t i v e mode ls fo r d r i l l i n g t h ru s t and to rque
I t i s shown th a t the 'Pl ane FSank' model ignor es
From these i n t r i c a t e ana lyse s , s imple r empi r i ca l
1.
2 .
3 .
4 .
5.
6.
7 .
8.
9 .
10.
11.
1 2 .
1 3 .
1 4 .
15 .
REFERENCES
M.E. MERCHANT, I . E . Aust . I nt . Conf . Prod. Tech. ,
Melbourne ( 1974) .
R .
TOURRET, " Perfor mance of Metal C ut ti ne Tools ".
-
Butteworth, London, ( 1 9 5 8 ) .
Machining Data Handbook, 3r d Ed. , Metcut Research
Assoc i a t e s Inc . , C inc inna t i , Ohio, ( 1 9 8 0 ) .
E.J.A.
ARMAREGO,
UNESCO-CIRP se min ar on Manuf.
Technology, Singapore, ( 1 9 7 2 ) .
A.S.T.M.E. 'To ol Eng ine ers Handb ook', McGraw
H i l l
New York ( 1 9 5 8 ) .
METAL CUTTING TOOL INSTITUTE, "Metal C ut ti ng Tool
Handbook ( 1969) .
AMERICAN STANDARD, USAS. B94- 11 - 1967 .
AUSTRALIAN STANDARD, AS 2438- 1981 .
E.J.A. ARMAREGO
and A. ROTENBERG, I n t . 3 Mach.
Tool Des. Res..
1 3 .
155. 165 and
183 ( 1973) .
E.J .A. ARMAREGO iiiid J.D: WRIGHT, Annais CIkp, 3
5 ( 1980) .
S .
WIRIYACOSOL and E . J .A .
ARMAREGO,
Annals CIRP,
2 8 , 8 7 ( 1 9 7 9 ) .
KALDOR and E. LENZ, Annals C I Y ,
2 9 , 2 3 ( 1 9 8 0 ) .
E . J . A .
ARMAREGO an d R.H. BROWN, TheTach in ing o f
Me ta l s" , Pren t i ce H a l l I n c . , New J e r s e y ( 1969) .
J . D . WRIGHT
and E.J .A.
ARMAREGO,
Annals. CIRP, 2
1 ( 1 9 8 3 ) .
J . D . WRIGHT, Ph.D. Th es is , U ni ve rs it y of Melbourne
( 1 9 8 1 ) .
NOMENCLATURE
b
-
wid th o f cu t
C C
- conica l gr inding method
parameters
Cx yC
C t 0 - s p e c i fi e d l i p c l ea r an c e a gg le a t t h e d r i l l
D,D'- nominal dr i l l d iameter and ch is e l edge diameter
- t o t a l c h i s e l e d g e f o r c e s p e r u n i t w i d th o f
lp
gut along and normal to
V
i n t h e v e l o c i t y p l a ne
pe r iphe ry
- t o t a l ch i se l edge fo rce s a long and norma l
i n t h e v e l o c i t y p l a ne
K
,K
-
edge fo rce s pe r un i t wid th o f c u t
a t
t h e
lip
'Qlong and perpe ndicul ar to Vw i n t h e p l a ne
L - ch i se l edge l eng th (=
D ' =
2 r )
Mi,Mc - se l e c t ed number o f e l ement sca t d r i l l l i p and
2p - s p e c i f i e d d r i l l p o i n t a n g le
r
- r ad ius a t t he mid-poin t o f an e lementa l cu t t i ng
r -
c h ip l e n g t h r a t i o
r
tL
- c u t t h i c k ne s s
n or ma l t o t h e l i p
ch i se l edge
edge
-
r ad ius a t t he ch i se l edge when anD
= 0
~ __
Tc,T,,Tt - t o rque a t t h e c h i s e l ed g e, l i p s an d t h e
w h o l e d r i l l
Thc,Th,,Th - t h r u s t a t t h e c h i s e l e dg e, l i p s a nd t h e
V e , V f , V w
- r e s u l t a n t , f e e d an d t a n g e n t ia l v e l o c i t i e s ,
2W - web thickness a t t h e d r i l l p oi n t
un,unD
3 , y n D
2ref
t wh ol e d r i l l
r e s p e c t i v e l y
- s t a t i c and dynamic normal c leara nce angles ,
r e s p e c t i v e l y
-
s t a t i c and dynamic normal rake an gle
- r e fe rence rake ang le a t t h e l i p s
-
ch is e l edge wedge angle
YW
9
-
8/11/2019 Predictive Models for Drillin-g Thrust and Torque - A Comparison of Three Flank
6/6
6 ,
6 0
-
hel i x angl es at any poi nt on t he l i p and at
oA, Ab - el ement al ar ea and w dth of cut
cl assi cal obl i ue cut t i ng f orce compon-
ent s due to dej ormati on at t he dr i l l
l i p el ement s
- cl assi cal ort hogonal cut t i ng force com
ponents due t o def or mat i on and edge
ef f ects at t he chi sel edge el ement s
-
cl assi cal obl i que cutt i ng f orce
dri l l l i p el ements
- Thr ust and tor que at t he kt h el ement at
t he chi sel edge
and edge ef f ects on the j t d l i p el ement
out er corner
A F ~ , A F ~ , A F ~
nFpC. AFFqC
AFpE, AF
os
-
el ement al cut t i ng edge l ength
AThck, ATck
AT
.,cThtj
-
t orques and t hrust due to ef ormat i on
- r efer ence angl e at t he l i ps
nc
a - gr i ndi ng cone sem - cone angl e
x - f r i ct i on angl e on the rake face
? ~B x 1 * ~2 n o r ma lr i ct i on angl e i n pl ane normal t o
QE' AFREcomponents due to edge ef f ect s at t he
I J
n
- r esul t ant cutt i ng speed angl e
-
chi p fl ow angl e on the r ake face f or dr i l l
l i p el ement s
- gri ndi ng cone gener at or pl an angl es [9]
- st at i c and dynam c angl es of i ncl i nat i on,
cutt i ng edge
r e spec i vel y
s
sD
T
- shear st r ess i n shear pl ane
-
normal shear angl es
z?w',wo
- dr i l l web angl es at any poi nt on t he l i p,
t he chi sel edge cor ner and the out er cor ner,
respect i vel y
X
-
coni cal gri ndi ng method par amet er
I
J'
-
dri l l speci f i ed chi sel edge angl e and i ts
compl ement , r especti vel y.
10