predictive models for drillin-g thrust and torque - a comparison of three flank

8/11/2019 Predictive Models for Drillin-g Thrust and Torque - A Comparison of Three Flank

1/6

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


2/6

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)

6


3/6

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


4/6

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 (


5/6

._

- - -

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


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

predictive models for drillin-g thrust and torque - a comparison of three flank

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