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Drilling model for cutting lip and chisel edge and comparison of experimental and predicted results. II — revised cutting lip model
Int. J. Much, Tool Des. Res. Printed in Great Britain
Vol. 25, No. 4. pp. 367-376. 1985.
002(~-7357/8553.(X)+.00 P e r g a m o n Press L i d
D R I L L I N G M O D E L F O R C U T T I N G LIP A N D C H I S E L E D G E AND COMPARISON OF E X P E R I M E N T A L AND PREDICTED RESULTS. II - R E V I S E D C U T T I N G LIP M O D E L A. R. WATSON* (Received 10 January 1985) Abstract--The previous model assumed a series of elements across the cutting lip and underestimated the torque and thrust, and is modified here to account for the necessary integrity of the chip. Because the integral chip flows initially as a unit in one direction and is rotating about a point, there must be a linear variation in the chip velocity across the lip. This variation in chip velocity effects the shear angle across the lip, effectively reducing the range from that calculated for a series of elements. When this extra work of effectively holding the various elements together was taken into account, predictions much closer to the experimental results were obtained. The modified model accounts reasonably for variation of most of the drill and process variables.
1.
INTRODUCTION
AN INITIAL drilling model [1], which had been developed assuming that the material machined by each cutting lip could be considered as a series of independent elements did not predict results that agreed with the experimental results. The initial assumptions. particularly the assumption that the material could be treated as a series of independent elements, were re-examined. The chip observed in drilling is complete across the cutting lip and there may be some restriction on the variation of the chip flow angle across the cutting lip because of the integral chip. There may also be, for the same reason, other variables involved in the chip formation process of an integral chip that may also be restricted in their range of values across the cutting lip in comparison to the range in values of the variables for a series of elements across the same width of cutting edge. 2.
CONSEQUENCES OF RIGID BODY MOTION OF CHIP
The actual chip produced on each cutting lip is integral along the length of the cutting edge and of constant cross-section along its length, and it will be shown that the initial chip flow angle is constant across the width of the chip along the cutting edge. It is generally assumed, and it is assumed here that the cutting edge is straight, and so lines in the rake face parallel to the edge will also be straight. The chip ribbon is of constant cross-section along its length, so that the intercepts by lines parallel to the cutting edge will all be of constant size and equal to the chip length along the cutting edge. It can thus be shown that when the chip ribbon flows along the rake face, the angles of the normal to the cutting edge in the rake face at each side of the chip ribbon are equal, so that the chip flow angle is constant across the cutting edge. Across the cutting edge there is a variation in velocity of the work relative to the cutting edge, a variation of the rake angle (Fig. 13 of [2]), and a small variation in the undeformed chip thickness (Fig. 4 of [1]), so that there may well be a variation in the thickness of the chip along the cutting lip, as well as a change in velocity of the chip along the cutting edge. The shear angle also may vary across the cutting lip. The chip, as a rigid body, can be either translating with a velocity that is constant for every point of the body or instantaneously rotating about a point. Because there is a variation in the chip velocity over the length of the cutting edge, the only possible motion of the chip is then an instantaneous rotation about a point. There will be, consequently, a linear variation in the chip velocity along the cutting edge and a * Department of Mechanical Engineering, Faculty of Military Studies, The University of New South Wales. Duntroon, ACT, Australia, 2600.
~,~Y/)R 53:4-~,'
367
368
A . R . WATSON
constancy of direction of the chip velocity along the cutting edge, i.e. the chip flow angle is constant along the cutting edge. There can be quite a large variation in the cutting velocity across the cutting lip, and it is unlikely that the shear angle will remain constant over that range considering that the rake angle also varies (Fig. 13 of [2]). In orthogonal machining, the chip velocity V is related to U, the velocity of the work relative to the tool, by the relationship
V-
U sin +
(1)
cos (6 - ~)
where d~ is the shear angle and ~/the rake angle. This relationship can be rearranged to yield, in the normal plane tan 4~ =
cos ~.e Ut
(2)
sin "~ne
V'
where U' is the velocity of the work relative to the tool in the normal plane and V' is the chip velocity in the normal plane. The variables ~/,,~, U' and V' change with the dimensionless radius term p, 2/"
P= D
(3)
(where r is the radius being considered and D is the drill dia.) across the cutting edge, so that if the forms of these changes are known, then the variation of d~n across the cutting edge can be determined. In [1] the working normal rake angle %e at any point on the lip was given in equation (21), and the variation of ~/,,e across the lip is shown in Fig. 13 of [2]. The cutting velocity U' will vary approximately linearly across the lip (if the variation in hse is neglected, refer to equation 18 of [1] and Fig. 13 of [2]) being almost zero at the centre of the drill (p = 0) and U0 at the outside radius (p = 1), so that U' would be given by a relationship of the form U' = pUo.
(4)
It has been shown that the actual chip velocity must vary linearly across the lip, and because the chip flows at a constant chip flow angle across the width of the chip, the normal component of the chip velocity must also vary linearly across the lip, so that in general V' = Vo ( ( l - A ) p + A)
(5)
where V0 is the normal component of the chip velocity at p = 1 and A V0 is the normal component of the chip velocity at p = 0. A is likely to be very small, so that to reasonable accuracy V' would be given by V' = pV0
(6)
The shear angle ~bn is thus given by tan d~, =
(7)
cos ~ne Uo
sin
"Yne
Vo
If the shear angle is known at any point across the cutting edge, then it would be
Drilling Model for Cutting Lip and Chisel Edge - - II
369
possible to determine the shear angle at any other value of p from equation (7) after the constant Uo/Vois found, because the variation of the working normal rake angle "Y,ewith p is known (equation (21) of [1]). There is thus some variation in the shear angle across the cutting edge. The variation in the normal shear angle across the cutting lip using the segmental approach for one set of conditions is shown in Fig. 1. If the normal shear angle at the outside radius for the integral chip is selected to be the same as the normal shear angle at the outside radius from the segmental approach, then using the value for the normal rake 7,,,, from Fig. 13 of [2], the quantity U./V(~can be determined from U0
B
Vo
COS "~ne
tan +,,
+ sin
~lne
(8)
•
When this value of Uo/Vois used with the values of ~,e from Fig. 13 of [2] in equation (7), the variation of the normal shear angle for the integral chip across the cutting lip is obtained, and is shown in Fig. 1. It will thus be seen that there is less variation in the normal shear angle across the lip when a solid chip is considered than if a series of elements is considered. In the segmental model, the shear planes of the individual elements would form a shear surface which has more twist down its length along the cutting edge than does the shear surface for the integral chip.
50~
Integral
40--
mental 30-
D = 1 5 - 8 8 mm K=60 =
20--
f =0.13 m m / r N =572 r/min
(~r)n
(-)
I0 1
0
I
'2
I
.4
I
I
"6
DIMENSIONLESS
'8
I'0
RADIUS.
FIG. l. Variation of normal shear angle across cutting lip for integral and segmental chip.
370
A . R . WATSON
If the chip velocity or the normal shear angle could be measured at any point on the cutting lip, then the constant Uo/Vocould be found (since the cutting velocity and the working normal rake angle 7,,e can be readily found at that point), and it would be possible to determine the variation in the normal shear angle for the integral chip across the cutting lip. Neither the normal shear angle nor the chip velocity are easily measured at any radius, even at the outside radius (p = 1), so it would be extremely difficult to determine experimentally an appropriate value of Uo/Vo. It would also be hard to measure the constant chip flow angle. Because of these experimental difficulties, other techniques, discussed in the next section, must be used to obtain appropriate values of Uo/Vo and the constant chip flow angle "qc~. The drilling model will be modified in the next section to take account of the constant chip flow angle and the linear variation of the chip velocity along the cutting edge for the integral chip. 3.
REVISION OF DRILLING MODEL FOR CUTTING LIP
As identified in the previous section, to account for the actual behaviour of the chip along the cutting lip, it is necessary to constrain all of the material to flow at the same chip flow angle and to obtain a linear variation in the chip velocity over the lip. It is likely that there will be a point along the cutting lip where the shear angle for the integral chip will correspond to the shear angle from the segmental approach at the same radius. The approach that has been adopted has been to assume that at radius p = Ps, the shear angle for the integral chip will be the same as for the segmental chip, and to use the P R E D I C T routine (of the machining model of Hastings [3] and further modified by Oxley and others [4-8]) to find the shear angle at that radius ps. Because the normal rake angle %e is determinable at any radius, the constant Uo/Vo can then be found from equation (8). For any element, the shear angle can then be found from equation (7), and so the cutting forces (as will be shown later) can be obtained. The torque and thrust contributions can also then be determined using the common chip flow angle rice. The radius p~ and the common chip flow angle ~qcc are unknowns, and these have been selected to bring the predicted results into closer correspondence with the experimental results. In the modified LIP programme (MLIP [9]), the material flowing onto the cutting lip is again considered as a series of elements. The normal shear angle qbn~ at radius Ps is obtained from a modified P R E D I C T subroutine. The constant Uo/Vocan then be found. using equation (8). The material is then considered as a series of elements, but now the normal shear angle for each element is found from equation (7), using the value of U~¢Vo just determined. This normal shear angle for each element is returned to the beginning of the DO loop in the P R E D I C T subroutine in the MLIP programme for one part-pass through the loop to determine the forces Fc and Fr, and the normal friction angle hn. On this part-pass through the DO loop no attempt is made to equate the shear stresses "rZNT and kcHzP. (In the P R E D I C T routine ([3-8]), the shear angle solution is obtained when the interface shear stress, as calculated from the shear zone forces ('rtNr), is equal to the shear stress obtained from a separate consideration of the interface conditions of strain rate and temperature (kcme).) The shear angle for an element of the integral chip will then not produce coincidence of these interface shear stresses, "QNTand kcHlP. In the MLIP programme, as in the LIP programme [1], the solution is obtained in the normal plane, so that F'c and F'r are determined (these normal plane forces F'c and F~- are equivalent to the forces Fc and F r respectively in the above) as well as the friction force F' and the friction angle k, in the normal plane. The friction force in the rake face and parallel to the cutting edge, Ff, now becomes Ff = sin h,, tan Xlcc V/F '2 + F'er
(9)
where "q~, is the common chip flow angle for the elements. Ff is perpendicular to F~ and F~ and forms a right handed cartesian set with F~ and F~-. The f o r c e s / ~ , F~- and Ff are in the same orientation as the corresponding forces in the
Drilling
Model
l’or Cutting
Lip and Chisel Edge -
II
371
previous derivation used in the LIP programme, and consequently the same equations for the increments of torque and thrust, viz. equations (27) and (28) of [l], can be used. The total torque and thrust result from the increments on the elements from the pilot hole edge to the outside dia. on the two cutting edges, and are given by equations (29) and (30) of [l] respectively. The modification effectively adds two new unknowns ps and qCC,and by varying these parameters it was found that when ps = 0.2 and qCC= - 3Y,
(10) (11)
good agreement between predicted and experimental results is obtained. For this value of pr, the variation of normal shear angle across the cutting lip for the integral chip is shown in Fig. 2 together with the variation of the shear angle from the segmental approach. It will be seen that there is much less variation for the integral chip than for the segmental chip, and that from p = 0.4 to p = 1.0 +, is nearly constant. This near constancy of shear angle over this range is the likely reason why previous approaches using average conditions over this region have produced reasonable results (section 4.3 of [ll). 50
40
30
lnlegral
2c
D =15.88mm K =600
+”
f =0.13
1’)
N -572
mm/r r/min
I(
I .2
I
.4 9 DIMENSIONLESS
FIG. 2. L’ariation
of normal
I
I .6
.8
0
RADIUS.
shear angle across cutting lip for integral
and segmental
chip.
372
A . R . WATSON
The predictions from the MLIP programme when the feed, speed, point angle, pilot hole dia. and relief angles are varied, are shown in Figs. 3-8 inclusive, together with the experimental results. Most of the predictions are now within the band of scatter of the experimental results or much closer to the experimental points than were obtained previously. Even the variable point angle predictions (Fig. 7), which may be expected to vary from the experimental points because of the departure from an assumed straight cutting edge of the actual drills when the point angle varies from 120 ° , fall much closer to the experimental points. The actual curvature of the edge will change the cutting force component in the tangential direction more than it will vary the axial cutting force component and so it is to be expected that the torque prediction will depart from the experimental results more than the thrust predictions will as the half point angle departs
x TORQUE 40 PREDICTED. 9S= constant.
trend line.
30 ?s variable
with feed
- * 6 % bond about trend line "~Z uJ 20 0 o: 0 p.
K-= 6 0 =
~1=
8*
~ 2 = 18 ° SPEED =572 r / r a i n
I0
PILOT HOLE DIAMETER = 2 mm DRILL DIAMETER = 15.88 mm PREDICTIONS FROM MODIFIED LIP PROGRAMME.
O
O.I
0"2
0'3 FEED
O,4
0.5
0-6
(ram/r)
FIG. 3. Effect of feed on torque when drilling with a pilot hole.
0.5
1.0
1.5
2-0
2.5
I O. I
I 0.2
PREDICTIONS
FEED
I 0.3
I 0.4
LIP
(ram/r)
FROM MODIFIED
15.88 mm
I 0.5
PROGRAMME
DIAMETER
DIAMETER= 2 mm DRILL
PILOT HOLE
SPEED=572 r/rain
g2 = 18"
.QI = 8 *
K = 60"
I 0.6
+-10% band about trend line
trend line
Illl~J" IL, I~X
FIG, 4. Effect of feed on thrust w h e n drilling with a pilot hole.
(/I 3 n"r
I-
9s v a r i a b l e with feed
Predicted
Predicted ?S: constant
x Thrust
tJJ 3
Z v
- - 0iv. 0 I.-
-
I 500
PREDICTIONS FROM MODIFIED LIP PROGRAMM
SPEED
I I000
line.
I
--
( r/min
).
I 1500
D I A M E T E R = 1 5 . 8 8 mm
D I A M E T E R = 2 mm DRILL
PILOT HOLE
FEED =0.13 m m / r
n 3 = 18 =
£/'1 = 8 °
I< = 6 0 °
trend
,r.n°,,n.
0.5
I'O
ne 2: I-
I-
Z
FIG. 5. Effect of speed on t o r q u e and thrust w h e n drilling with a pilot hole.
0
IO
.
~" I 0 % b a n d a b o u t trend line,
J
e~
r-
c"3
e'~
0~ EO
374
A . R . WATSON
×
40
n
-4 "///'~J(//J////~
THRUST
I¢ - 6 0
X.
1.0
(=
rEED.,, ....
E
°
,P,ED
=,,,
ram,, ,,.,o Z
Z
-'//J~'//.
" / / / _ ~ experlmenta I
U.I '-t
I-
0 0 I-
"I"
I-
5-
--
.
O
5
,
,..
I0
15
PILOT HOLE DIAMETER
0.5
(ram)
Fro. 6. Effect of pilot hole dia. on torque and thrust.
from 60 °. The predicted torque and thrust results (Figs. 3 and 4) can be seen to depart from the band of experimental points at the higher feeds. If, however, p~ is allowed to vary with feed, a much better correspondence between predicted and experimental points can be obtained. Figs. 3 and 4 show the predicted response when p~ varies linearly with feed, viz
(12)
p, = 0.193 + 0.054 f
where f i s the feed in mm/r. This changes Ps from Ps = 0.2 at a feed of 0.13 mm/r to Ps = 0.22 for a feed of 0.5 mm/r, and even though this change is small, it is sufficient to give the better agreement. The value of "qccof - 3 5 ° does not correspond to any of the chip flow relationships at p = ps, for, as can be seen from Fig. 13 of [2], hse is about +30 ° at this radius. The sense and magnitude of the chip flow direction must be related to the instantaneous rotation of the chip. The whole chip formation process must be related to the geometry of the cutting lip and the instantaneous rotation of the chip, because the point of shear angle coincidence remains constant at p = 0.2 even for the pilot hole tests when, for most of the range of holes drilled, p = 0.2 falls outside the material being machined. 4.
CONCLUSIONS
When due account is taken of the integral motion of the chip from the cutting lip, then predictions of the torque and thrust, when drilling with a pilot hole that removes the material that would be machined by the chisel edge, can be obtained that agree reasonably with experimental results when the drill and process variables are changed.
I
30
PREDICTIONS
0 I--
P
R
/ E D I C T E D
PRED,CTED
X
I 50
HALF POINT ANGLE
I 40
(°)
I 60
X
Q
I 70
THRUST
ORQOE
X
FROM MODIFIED LIP PROGRAMME.
/
~
0 /
x
I 80
0.5
1.0
Fie,. 7. Effect of p o i n t angle on t o r q u e a n d t h r u s t w h e n d r i l l i n g w i t h a p i l o t hole.
0
2.-
4
12 -
= 3mm =i5.88mm =0.13 m m / r =572 r / rain
/2 =18 ° Pilot Hole
Diameter Drill Diameter Feed ~,~. Speed
,0. : 8 °
x TORQUE
a THRUST
0
I
• •
+
--
SECONDARY
I I0
RELIEF
PREDICTIONS FROM MODIFIED L i P
ANGLE
.C~= Io )
( 20
PROGRAMME.
Diameter = 3ram Drill D i a m e t e r = 1 5 . 8 8 m m
•
..-11
~t = 18°
~'a = 18° O'X 0x
K =60" Speed = 5 7 2 r / r a i n Feed =0.13 m m / r Pilot Hole
PREDICTED TORQUE
=r/.,= 8 ~ ,
THRUST
PREDICTED
~'~1 = 8°
Gn=I8 °
~t = 8 °
PREDICTED~ ~ TORQUE÷ ~,.-2"
a ~= 2 °
÷ o
B°
~t= ~t=18=
2o a
THRUST ~,=
0.5
l-
I-(t) n"r
z
A
- I'O
FIG. 8. Effect of relief a n g l e s o n t o r q u e a n d t h r u s t w h e n d r i l l i n g w i t h a pilot hole.
,-i O fit: O I--
Z
IO--
15-
x
TORQUE ~n = 2 °
I
t-"
_=
¢-
o
rl~
376
A . R . WATSON
REFERENCES [1] A. R. WATSON,Int. J. Mach. Tool Des. Res. 2,$, 347 (1985). [2] A. R. WATSON,Int. J. Mach. Tool Des. Res. 25, 209 (1985). [3] W. F. HASTINGS,Theoretical and experimental investigation of the machining process, Ph.D. Thesis, University of New South Wales (1975). [4] P. L. B. OXLEYand W. F. HASnNOS, Phil. Trans. R. Soc. Lon. A. 282, 565 (1976). [5] P. L. B. OXLEVand W. F. HASrlNt;s, Proc. R. Soc. Lond. A. 356, 395 (1977). [6] P. MATHEW,W. F. HASl"INCSand P. L. B. OxI.EY, Proc. 2nd Conf. Mechanical Properties of Materials at High Rates of Strain, pp. 360--371. Institute of Physics, Bristol (1979). [7] W. F. H^S'aNGS, P. MATn~Wand P. L. B. OXLEV, Proc. R. Soc. Lond. A. 371,569 (1980). [8] P. MArnEW and P. L. B. OxI.EY, Wear 69, 219 (1981). [9] A. R. WATSON,Theoretical prediction of torque and thrust in drilling, Ph.D. Thesis, University of New South Wales (1984).
Vol. 25, No. 4. pp. 367-376. 1985.
002(~-7357/8553.(X)+.00 P e r g a m o n Press L i d
D R I L L I N G M O D E L F O R C U T T I N G LIP A N D C H I S E L E D G E AND COMPARISON OF E X P E R I M E N T A L AND PREDICTED RESULTS. II - R E V I S E D C U T T I N G LIP M O D E L A. R. WATSON* (Received 10 January 1985) Abstract--The previous model assumed a series of elements across the cutting lip and underestimated the torque and thrust, and is modified here to account for the necessary integrity of the chip. Because the integral chip flows initially as a unit in one direction and is rotating about a point, there must be a linear variation in the chip velocity across the lip. This variation in chip velocity effects the shear angle across the lip, effectively reducing the range from that calculated for a series of elements. When this extra work of effectively holding the various elements together was taken into account, predictions much closer to the experimental results were obtained. The modified model accounts reasonably for variation of most of the drill and process variables.
1.
INTRODUCTION
AN INITIAL drilling model [1], which had been developed assuming that the material machined by each cutting lip could be considered as a series of independent elements did not predict results that agreed with the experimental results. The initial assumptions. particularly the assumption that the material could be treated as a series of independent elements, were re-examined. The chip observed in drilling is complete across the cutting lip and there may be some restriction on the variation of the chip flow angle across the cutting lip because of the integral chip. There may also be, for the same reason, other variables involved in the chip formation process of an integral chip that may also be restricted in their range of values across the cutting lip in comparison to the range in values of the variables for a series of elements across the same width of cutting edge. 2.
CONSEQUENCES OF RIGID BODY MOTION OF CHIP
The actual chip produced on each cutting lip is integral along the length of the cutting edge and of constant cross-section along its length, and it will be shown that the initial chip flow angle is constant across the width of the chip along the cutting edge. It is generally assumed, and it is assumed here that the cutting edge is straight, and so lines in the rake face parallel to the edge will also be straight. The chip ribbon is of constant cross-section along its length, so that the intercepts by lines parallel to the cutting edge will all be of constant size and equal to the chip length along the cutting edge. It can thus be shown that when the chip ribbon flows along the rake face, the angles of the normal to the cutting edge in the rake face at each side of the chip ribbon are equal, so that the chip flow angle is constant across the cutting edge. Across the cutting edge there is a variation in velocity of the work relative to the cutting edge, a variation of the rake angle (Fig. 13 of [2]), and a small variation in the undeformed chip thickness (Fig. 4 of [1]), so that there may well be a variation in the thickness of the chip along the cutting lip, as well as a change in velocity of the chip along the cutting edge. The shear angle also may vary across the cutting lip. The chip, as a rigid body, can be either translating with a velocity that is constant for every point of the body or instantaneously rotating about a point. Because there is a variation in the chip velocity over the length of the cutting edge, the only possible motion of the chip is then an instantaneous rotation about a point. There will be, consequently, a linear variation in the chip velocity along the cutting edge and a * Department of Mechanical Engineering, Faculty of Military Studies, The University of New South Wales. Duntroon, ACT, Australia, 2600.
~,~Y/)R 53:4-~,'
367
368
A . R . WATSON
constancy of direction of the chip velocity along the cutting edge, i.e. the chip flow angle is constant along the cutting edge. There can be quite a large variation in the cutting velocity across the cutting lip, and it is unlikely that the shear angle will remain constant over that range considering that the rake angle also varies (Fig. 13 of [2]). In orthogonal machining, the chip velocity V is related to U, the velocity of the work relative to the tool, by the relationship
V-
U sin +
(1)
cos (6 - ~)
where d~ is the shear angle and ~/the rake angle. This relationship can be rearranged to yield, in the normal plane tan 4~ =
cos ~.e Ut
(2)
sin "~ne
V'
where U' is the velocity of the work relative to the tool in the normal plane and V' is the chip velocity in the normal plane. The variables ~/,,~, U' and V' change with the dimensionless radius term p, 2/"
P= D
(3)
(where r is the radius being considered and D is the drill dia.) across the cutting edge, so that if the forms of these changes are known, then the variation of d~n across the cutting edge can be determined. In [1] the working normal rake angle %e at any point on the lip was given in equation (21), and the variation of ~/,,e across the lip is shown in Fig. 13 of [2]. The cutting velocity U' will vary approximately linearly across the lip (if the variation in hse is neglected, refer to equation 18 of [1] and Fig. 13 of [2]) being almost zero at the centre of the drill (p = 0) and U0 at the outside radius (p = 1), so that U' would be given by a relationship of the form U' = pUo.
(4)
It has been shown that the actual chip velocity must vary linearly across the lip, and because the chip flows at a constant chip flow angle across the width of the chip, the normal component of the chip velocity must also vary linearly across the lip, so that in general V' = Vo ( ( l - A ) p + A)
(5)
where V0 is the normal component of the chip velocity at p = 1 and A V0 is the normal component of the chip velocity at p = 0. A is likely to be very small, so that to reasonable accuracy V' would be given by V' = pV0
(6)
The shear angle ~bn is thus given by tan d~, =
(7)
cos ~ne Uo
sin
"Yne
Vo
If the shear angle is known at any point across the cutting edge, then it would be
Drilling Model for Cutting Lip and Chisel Edge - - II
369
possible to determine the shear angle at any other value of p from equation (7) after the constant Uo/Vois found, because the variation of the working normal rake angle "Y,ewith p is known (equation (21) of [1]). There is thus some variation in the shear angle across the cutting edge. The variation in the normal shear angle across the cutting lip using the segmental approach for one set of conditions is shown in Fig. 1. If the normal shear angle at the outside radius for the integral chip is selected to be the same as the normal shear angle at the outside radius from the segmental approach, then using the value for the normal rake 7,,,, from Fig. 13 of [2], the quantity U./V(~can be determined from U0
B
Vo
COS "~ne
tan +,,
+ sin
~lne
(8)
•
When this value of Uo/Vois used with the values of ~,e from Fig. 13 of [2] in equation (7), the variation of the normal shear angle for the integral chip across the cutting lip is obtained, and is shown in Fig. 1. It will thus be seen that there is less variation in the normal shear angle across the lip when a solid chip is considered than if a series of elements is considered. In the segmental model, the shear planes of the individual elements would form a shear surface which has more twist down its length along the cutting edge than does the shear surface for the integral chip.
50~
Integral
40--
mental 30-
D = 1 5 - 8 8 mm K=60 =
20--
f =0.13 m m / r N =572 r/min
(~r)n
(-)
I0 1
0
I
'2
I
.4
I
I
"6
DIMENSIONLESS
'8
I'0
RADIUS.
FIG. l. Variation of normal shear angle across cutting lip for integral and segmental chip.
370
A . R . WATSON
If the chip velocity or the normal shear angle could be measured at any point on the cutting lip, then the constant Uo/Vocould be found (since the cutting velocity and the working normal rake angle 7,,e can be readily found at that point), and it would be possible to determine the variation in the normal shear angle for the integral chip across the cutting lip. Neither the normal shear angle nor the chip velocity are easily measured at any radius, even at the outside radius (p = 1), so it would be extremely difficult to determine experimentally an appropriate value of Uo/Vo. It would also be hard to measure the constant chip flow angle. Because of these experimental difficulties, other techniques, discussed in the next section, must be used to obtain appropriate values of Uo/Vo and the constant chip flow angle "qc~. The drilling model will be modified in the next section to take account of the constant chip flow angle and the linear variation of the chip velocity along the cutting edge for the integral chip. 3.
REVISION OF DRILLING MODEL FOR CUTTING LIP
As identified in the previous section, to account for the actual behaviour of the chip along the cutting lip, it is necessary to constrain all of the material to flow at the same chip flow angle and to obtain a linear variation in the chip velocity over the lip. It is likely that there will be a point along the cutting lip where the shear angle for the integral chip will correspond to the shear angle from the segmental approach at the same radius. The approach that has been adopted has been to assume that at radius p = Ps, the shear angle for the integral chip will be the same as for the segmental chip, and to use the P R E D I C T routine (of the machining model of Hastings [3] and further modified by Oxley and others [4-8]) to find the shear angle at that radius ps. Because the normal rake angle %e is determinable at any radius, the constant Uo/Vo can then be found from equation (8). For any element, the shear angle can then be found from equation (7), and so the cutting forces (as will be shown later) can be obtained. The torque and thrust contributions can also then be determined using the common chip flow angle rice. The radius p~ and the common chip flow angle ~qcc are unknowns, and these have been selected to bring the predicted results into closer correspondence with the experimental results. In the modified LIP programme (MLIP [9]), the material flowing onto the cutting lip is again considered as a series of elements. The normal shear angle qbn~ at radius Ps is obtained from a modified P R E D I C T subroutine. The constant Uo/Vocan then be found. using equation (8). The material is then considered as a series of elements, but now the normal shear angle for each element is found from equation (7), using the value of U~¢Vo just determined. This normal shear angle for each element is returned to the beginning of the DO loop in the P R E D I C T subroutine in the MLIP programme for one part-pass through the loop to determine the forces Fc and Fr, and the normal friction angle hn. On this part-pass through the DO loop no attempt is made to equate the shear stresses "rZNT and kcHzP. (In the P R E D I C T routine ([3-8]), the shear angle solution is obtained when the interface shear stress, as calculated from the shear zone forces ('rtNr), is equal to the shear stress obtained from a separate consideration of the interface conditions of strain rate and temperature (kcme).) The shear angle for an element of the integral chip will then not produce coincidence of these interface shear stresses, "QNTand kcHlP. In the MLIP programme, as in the LIP programme [1], the solution is obtained in the normal plane, so that F'c and F'r are determined (these normal plane forces F'c and F~- are equivalent to the forces Fc and F r respectively in the above) as well as the friction force F' and the friction angle k, in the normal plane. The friction force in the rake face and parallel to the cutting edge, Ff, now becomes Ff = sin h,, tan Xlcc V/F '2 + F'er
(9)
where "q~, is the common chip flow angle for the elements. Ff is perpendicular to F~ and F~ and forms a right handed cartesian set with F~ and F~-. The f o r c e s / ~ , F~- and Ff are in the same orientation as the corresponding forces in the
Drilling
Model
l’or Cutting
Lip and Chisel Edge -
II
371
previous derivation used in the LIP programme, and consequently the same equations for the increments of torque and thrust, viz. equations (27) and (28) of [l], can be used. The total torque and thrust result from the increments on the elements from the pilot hole edge to the outside dia. on the two cutting edges, and are given by equations (29) and (30) of [l] respectively. The modification effectively adds two new unknowns ps and qCC,and by varying these parameters it was found that when ps = 0.2 and qCC= - 3Y,
(10) (11)
good agreement between predicted and experimental results is obtained. For this value of pr, the variation of normal shear angle across the cutting lip for the integral chip is shown in Fig. 2 together with the variation of the shear angle from the segmental approach. It will be seen that there is much less variation for the integral chip than for the segmental chip, and that from p = 0.4 to p = 1.0 +, is nearly constant. This near constancy of shear angle over this range is the likely reason why previous approaches using average conditions over this region have produced reasonable results (section 4.3 of [ll). 50
40
30
lnlegral
2c
D =15.88mm K =600
+”
f =0.13
1’)
N -572
mm/r r/min
I(
I .2
I
.4 9 DIMENSIONLESS
FIG. 2. L’ariation
of normal
I
I .6
.8
0
RADIUS.
shear angle across cutting lip for integral
and segmental
chip.
372
A . R . WATSON
The predictions from the MLIP programme when the feed, speed, point angle, pilot hole dia. and relief angles are varied, are shown in Figs. 3-8 inclusive, together with the experimental results. Most of the predictions are now within the band of scatter of the experimental results or much closer to the experimental points than were obtained previously. Even the variable point angle predictions (Fig. 7), which may be expected to vary from the experimental points because of the departure from an assumed straight cutting edge of the actual drills when the point angle varies from 120 ° , fall much closer to the experimental points. The actual curvature of the edge will change the cutting force component in the tangential direction more than it will vary the axial cutting force component and so it is to be expected that the torque prediction will depart from the experimental results more than the thrust predictions will as the half point angle departs
x TORQUE 40 PREDICTED. 9S= constant.
trend line.
30 ?s variable
with feed
- * 6 % bond about trend line "~Z uJ 20 0 o: 0 p.
K-= 6 0 =
~1=
8*
~ 2 = 18 ° SPEED =572 r / r a i n
I0
PILOT HOLE DIAMETER = 2 mm DRILL DIAMETER = 15.88 mm PREDICTIONS FROM MODIFIED LIP PROGRAMME.
O
O.I
0"2
0'3 FEED
O,4
0.5
0-6
(ram/r)
FIG. 3. Effect of feed on torque when drilling with a pilot hole.
0.5
1.0
1.5
2-0
2.5
I O. I
I 0.2
PREDICTIONS
FEED
I 0.3
I 0.4
LIP
(ram/r)
FROM MODIFIED
15.88 mm
I 0.5
PROGRAMME
DIAMETER
DIAMETER= 2 mm DRILL
PILOT HOLE
SPEED=572 r/rain
g2 = 18"
.QI = 8 *
K = 60"
I 0.6
+-10% band about trend line
trend line
Illl~J" IL, I~X
FIG, 4. Effect of feed on thrust w h e n drilling with a pilot hole.
(/I 3 n"r
I-
9s v a r i a b l e with feed
Predicted
Predicted ?S: constant
x Thrust
tJJ 3
Z v
- - 0iv. 0 I.-
-
I 500
PREDICTIONS FROM MODIFIED LIP PROGRAMM
SPEED
I I000
line.
I
--
( r/min
).
I 1500
D I A M E T E R = 1 5 . 8 8 mm
D I A M E T E R = 2 mm DRILL
PILOT HOLE
FEED =0.13 m m / r
n 3 = 18 =
£/'1 = 8 °
I< = 6 0 °
trend
,r.n°,,n.
0.5
I'O
ne 2: I-
I-
Z
FIG. 5. Effect of speed on t o r q u e and thrust w h e n drilling with a pilot hole.
0
IO
.
~" I 0 % b a n d a b o u t trend line,
J
e~
r-
c"3
e'~
0~ EO
374
A . R . WATSON
×
40
n
-4 "///'~J(//J////~
THRUST
I¢ - 6 0
X.
1.0
(=
rEED.,, ....
E
°
,P,ED
=,,,
ram,, ,,.,o Z
Z
-'//J~'//.
" / / / _ ~ experlmenta I
U.I '-t
I-
0 0 I-
"I"
I-
5-
--
.
O
5
,
,..
I0
15
PILOT HOLE DIAMETER
0.5
(ram)
Fro. 6. Effect of pilot hole dia. on torque and thrust.
from 60 °. The predicted torque and thrust results (Figs. 3 and 4) can be seen to depart from the band of experimental points at the higher feeds. If, however, p~ is allowed to vary with feed, a much better correspondence between predicted and experimental points can be obtained. Figs. 3 and 4 show the predicted response when p~ varies linearly with feed, viz
(12)
p, = 0.193 + 0.054 f
where f i s the feed in mm/r. This changes Ps from Ps = 0.2 at a feed of 0.13 mm/r to Ps = 0.22 for a feed of 0.5 mm/r, and even though this change is small, it is sufficient to give the better agreement. The value of "qccof - 3 5 ° does not correspond to any of the chip flow relationships at p = ps, for, as can be seen from Fig. 13 of [2], hse is about +30 ° at this radius. The sense and magnitude of the chip flow direction must be related to the instantaneous rotation of the chip. The whole chip formation process must be related to the geometry of the cutting lip and the instantaneous rotation of the chip, because the point of shear angle coincidence remains constant at p = 0.2 even for the pilot hole tests when, for most of the range of holes drilled, p = 0.2 falls outside the material being machined. 4.
CONCLUSIONS
When due account is taken of the integral motion of the chip from the cutting lip, then predictions of the torque and thrust, when drilling with a pilot hole that removes the material that would be machined by the chisel edge, can be obtained that agree reasonably with experimental results when the drill and process variables are changed.
I
30
PREDICTIONS
0 I--
P
R
/ E D I C T E D
PRED,CTED
X
I 50
HALF POINT ANGLE
I 40
(°)
I 60
X
Q
I 70
THRUST
ORQOE
X
FROM MODIFIED LIP PROGRAMME.
/
~
0 /
x
I 80
0.5
1.0
Fie,. 7. Effect of p o i n t angle on t o r q u e a n d t h r u s t w h e n d r i l l i n g w i t h a p i l o t hole.
0
2.-
4
12 -
= 3mm =i5.88mm =0.13 m m / r =572 r / rain
/2 =18 ° Pilot Hole
Diameter Drill Diameter Feed ~,~. Speed
,0. : 8 °
x TORQUE
a THRUST
0
I
• •
+
--
SECONDARY
I I0
RELIEF
PREDICTIONS FROM MODIFIED L i P
ANGLE
.C~= Io )
( 20
PROGRAMME.
Diameter = 3ram Drill D i a m e t e r = 1 5 . 8 8 m m
•
..-11
~t = 18°
~'a = 18° O'X 0x
K =60" Speed = 5 7 2 r / r a i n Feed =0.13 m m / r Pilot Hole
PREDICTED TORQUE
=r/.,= 8 ~ ,
THRUST
PREDICTED
~'~1 = 8°
Gn=I8 °
~t = 8 °
PREDICTED~ ~ TORQUE÷ ~,.-2"
a ~= 2 °
÷ o
B°
~t= ~t=18=
2o a
THRUST ~,=
0.5
l-
I-(t) n"r
z
A
- I'O
FIG. 8. Effect of relief a n g l e s o n t o r q u e a n d t h r u s t w h e n d r i l l i n g w i t h a pilot hole.
,-i O fit: O I--
Z
IO--
15-
x
TORQUE ~n = 2 °
I
t-"
_=
¢-
o
rl~
376
A . R . WATSON
REFERENCES [1] A. R. WATSON,Int. J. Mach. Tool Des. Res. 2,$, 347 (1985). [2] A. R. WATSON,Int. J. Mach. Tool Des. Res. 25, 209 (1985). [3] W. F. HASTINGS,Theoretical and experimental investigation of the machining process, Ph.D. Thesis, University of New South Wales (1975). [4] P. L. B. OXLEYand W. F. HASnNOS, Phil. Trans. R. Soc. Lon. A. 282, 565 (1976). [5] P. L. B. OXLEVand W. F. HASrlNt;s, Proc. R. Soc. Lond. A. 356, 395 (1977). [6] P. MATHEW,W. F. HASl"INCSand P. L. B. OxI.EY, Proc. 2nd Conf. Mechanical Properties of Materials at High Rates of Strain, pp. 360--371. Institute of Physics, Bristol (1979). [7] W. F. H^S'aNGS, P. MATn~Wand P. L. B. OXLEV, Proc. R. Soc. Lond. A. 371,569 (1980). [8] P. MArnEW and P. L. B. OxI.EY, Wear 69, 219 (1981). [9] A. R. WATSON,Theoretical prediction of torque and thrust in drilling, Ph.D. Thesis, University of New South Wales (1984).