The mechanics of symmetrical circular form tool cutting

Int. J. Mach. Tool Des. Res.

Vol. 10, pp. 293-303.

Pergamon Press 1970. Printed in Great Britain

THE MECHANICS OF SYMMETRICAL CIRCULAR FORM TOOL CUTTING W. K. LUK*

(Received20 October 1969) Abstract--The analysis reveals the close relationship between the mechanics of orthogonal cutting and that of symmetrical circular form tool cutting. There is also a great similarity between symmetrical circular and symmetrical vee form tool cutting. INTRODUCTION AS DISCUSSED in an earlier paper [1], all practical cutting operations can be classified into three general groups, orthogonal, oblique and form tool cutting. Form tool cutting is defined as a process where there are two or more cutting edges cutting at the same time and their actions interact with each other. The mechanics of symmetrical vee form tool cutting has already been analysed [1]. The next logical step is to analyse the symmetrical circular form tool cutting. The combination of the vee and circular form tool cutting will cover most of the practical cutting operations. For a given rake angle, all symmetrical circular form tools are the same. The geometry of cut depends only on the ratio of T/r for full depth cuts, Fig. 1, and on the ratio t/r for non-full depth cuts, Fig. 5.

~n 28 28' S T , 81

A

Cn CeT ¢eTc ~etc 3"

Ao CT

kT kcv

NOMENCLATURE normal rake angle effective rake angle in the direction of chip flow included angle of a symmetrical vee form tool on the plane perpendicular to the cutting velocity included angle of a symmetrical vee form tool on the tool face angles on the plane perpendicular to the cutting velocity apparent friction angle normal shear angle effective shear angle in the direction of chip flow in symmetrical vee form tool cutting with full depth cuts effective shear angle in the direction of chip flow in symmetrical circular form tool cutting with full depth cuts effective shear angle in the direction of chip flow in symmetrical circular form tool cutting with non-full depth cuts maximum shear stress area of cut edge force per unit length in form tool cutting total specific cutting energy in form tool cutting with full depth cuts specific energy due to chip formation in form tool cutting with full depth cuts

* Manufacturing Science Laboratory, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada. 293

294

W . K . LUK

keg'

specific energy due to the edge force in form tool cutting with full depth cuts total specific cutting energy in form tool cutting with non-full depth cuts projection of the total length of engaged cutting edge on the plane perpendicular to the cutting velocity radius of the symmetrical circular form tool depth of cut in form tool cutting with non-full depth cuts depth of cut in form tool cutting with full depth cuts

kt L r t

T

ANALYSIS

1. Full depth cuts (cuts on a flat surface) (a) Shear angle. From Fig. 1, it can be seen that a circular cutting edge is equivalent to one consisting of an infinite pair of very small cutting edges joining together with each pair having a different value for the included angle 23.

FIG. 1. Full depth cuts.

The effective shear angle in symmetrical vee form tool cutting has been shown to be related to 8 by the following expression [1], cot SeT -- cosC°San~esin 8' cot 4'n -- (tan ae -- sin 3' cosSinan]ae/

(la)

where tan an = tan ae sin 8

(2)

tan 3' = tan 3 cos ae

(3)

and ~n = normal shear angle C~e = effective rake angle in the direction of chip flow an z normal rake angle 28' = included angle on the tool face. When

ae=0,3'=8,

and

an=0,

cot ~'er = sin 3 cot ~n-

(lb)

For a small element of cutting edge near the axis of a circular form tool, 3 is nearly 90 °, and

The Mechanics of Symmetrical Circular Form Tool Cutting

295

the effective shear angle becomes nearly the same as the shear angle in orthogonal cutting. For an element near B in Fig. 1, ~ approaches 3T, and CeT will be given by (la) or (lb). Therefore the effective shear angle in symmetrical circular form tool cutting with a depth of cut T is the mean value of the effective shear angles of all the small elements. It must lie between that in orthogonal cutting and that in symmetrical vee form tool cutting when First of all, take the simple case when ~ ----O. Equation (lb) can be rewritten as tanCeT--

tan Cn sin3"

One simple way to find the mean effective shear angle mean value of tan CeT. tan CeTc

If

½Ao

(lc)

~eTc for

the present case is to find the

tan Cef dA dT

1 f = ½Ao

tan Cn sin 3 dA

?iT

where Ao -~ area of cut. From Fig. 1, dA=

[r 2sin 2 3 - r ( r - T )

sin3]d3

½Ao = ¼rZ[2(½~ -- 3T) -- sin 23T] ½n f

tan Cn

sin~ dA

= r 2 tan Cn[COS 3T • tan CeTc -_

sin ~T(½~ -- 3T)]

r 2 tan ¢[cos 3T -- sin 3T(-½~r-- 3T)] ¼rZ[2--Q~- gT) -- sin-23-T] 4[cos 3T -- sin 3T(½~r-- 3T)] [2(½~ -- 3T) -- sin 2~T]

(4)

When ~e is not zero, (la) has to be used, and a simple expression for the mean effective shear angle similar to (4) will not be possible. Equations (la) and (lb) are plotted in Fig. 2 with the corresponding experimental points [1 ]. It shows that CeT/¢ in symmetrical vee form tool cutting does not change very much with ate and as a first approximation, equation (4) can also be used when Cte is not zero. This attempt is to sacrifice some accuracy to achieve simplicity• (b) The specific cutting energy, kT. The specific cutting energy, kT, is made up of two parts [2]. The first part, keT, is required to shear the work material at the shear plane and to overcome friction on the rake face of the tool. The second part, k~T, is associated with the bluntness of the cutting edges. The chip is assumed to be formed in the manner shown in Fig. 3. The shear plane extends from the circular cutting edge A D C to the work surface ABC.

296

W.K.

LUK

2

~eT

0 ~ 0

1.5

•

w

.5

_

•

20

T H E O R E T I C A L CURVE

I

o o E X P E R I M E N T A L POINTS

---

0

~

THEORETICAL CURVE • EXPERIMENTAL POINTS

I

I

50

40

I

50

(2 e = 0 °

J ct e = 22.10

I

60

I

70

80

FIG. 2.

CHIP--~

C

;

L_

O FIG. 3. Chip deformation model.

The shear plane is the curved surface ABCDA. The chip cross-section is the projection of ABCA on a plane perpendicular to the chip flow velocity. The same model as in symmetrical vee form tool cutting will be used, and k r will be given by the following expression [1], k T ~ ke~ + keT "r COS (A - - ae) s i n ~eTc COS (¢eTc -y h -- ae)

cTL + --Ao

(5)

where -r = maximum shear stress h = apparent friction angle c~- = edge force per unit length in symmetrical vee form tool cutting and assumed to be the same for circular form tool cutting L = projection of the total engaged cutting edge length on the plane perpendicular to the cutting velocity.

The Mechanics of Symmetrical Circular Form Tool Cutting

297

Lee and Shaffer's shear angle relation [3] ( a e T c ~ - ¼~r - -

(6)

)t - ~ o~e

is also assumed to hold [4]. (c) Chip cross-section and chip tool contact. The shapes of the chip cross-section and the resultant chip tool contact are shown in Fig. 4. They are constructed in the same manner as in the lathe turning operation [5].

E"

i

J427

CHIP-TOOL CONTACT

B"

/ C./

A,I CHIP CROSS- SECTION

ROSS-SECTION

Flo. 4. Full depth cuts, 3rd < projection.

0

,

FiG. 5. Non-full depth cuts.

/J

298

W.K. LUK

2. Non-full depth cuts

(a) Shear angle. The simplest non-full depth cut is when the circular form tool is making a depth of cut t on an existing circular groove having the same radius r as the tool. The following analysis will be confined to this simple case. Using the same approach as in full depth cuts, the mean effective shear angle ~etc when ~e = 0 is given by the following expression, Fig. 5,

if

tan Cetc -- ½Ao

tan CeT dA

o

dA = t d x = trd3 sin 3 ½Ao = ½(2rt) = rt ~½ 1 f tanCn tan Cetc = rt j ~ trd3 sin 3 0

½~ ----ranCh I- d3 0

= ½~ tan Cn = 1"57 tan Cn.

(7)

The above expression is again assumed to be valid when ae is not zero. When the ratio t/r decreases, the constraint imposed by each element on the rest of the chip also decreases. When t/r is small enough, each element will behave as if it is flowing freely by itself and the shear angle of each element will tend to have the same value as in orthogonal or oblique cutting. Therefore, when t/r decreases from large values to very small values, the effective shear angle of the whole chip will decrease from that given by (7) to that in orthogonal or oblique cutting. A simple empirical equation that satisfies the above and fits the experimental results is found to be given by the following expression:

¢.

¢.

It is assumed that Cac = CeTe when t = T, and the above expression is only valid between t/T---- 0.1 and t / T = 1. (b) The specific cutting energy, kt. It can be calculated in exactly the same manner as for full depth cuts. (c) Chip cross-section and chip tool contact. The shapes of the chip cross-section and the chip tool contact are shown in Fig. 6. E X P E R I M E N T A L R E S U L T S AND D I S C U S S I O N A three-component lathe dynamometer was secured to the ram of a milling machine by a special fixture. An aluminum workpiece was clamped on the milling vice which was in turn secured to the table, this being fed to the stationary tool by the table feed. The cutting speed used in all tests was 32 in./min. High-speed steel tools were used, and they were ground by

The Mechanics of Symmetrical Circular Form Tool Cutting

A'

"t

299

// B"

E"

CHIP-TOOL CONTACT

F" E'

IB ,

:?5,:: :"

:"

CHIP CROSS-SECTION

C"t: ~" C,I

IJ

¢_--B

A C

¢_ --

iD CROSS -SECTION OF CUT

~F FIG. 6. Non-full depth cuts, 3rd < projection.

means of a special radius grinding fixture. The radius of the circular form tools was measured accurately by means of an optical projector with a magnification of x 100. ;The depth of cut was determined by means of a tool m a k e r ' s microscope. Chip lengths were measured with a magnification × 10-20 by means of the optical projector. It was found that [1], in orthogonal cutting, when ~ = 0, ~ = 12.5, k = 219, and when = 22.1, ~b = 23-8, k = 124. The average value o f t is 36.8 × 103 psi. In form tool cutting, when ~e = 0, cT = 228 lb/in., and when ~e := 22"1, c~' = 178 lb/in. In oblique cutting, when ~ = 0, ~n = q~ = 12"5 °, and when an = 22"1 °, q~n = ~ = 23"8. Experimental results are c o m p a r e d with theoretical values for symmetrical circular f o r m tools of different rake angles and radii with full depth and non-full depth cuts in Figs. 7-13. The slightly lower experimental values of Fig. 9 are consistent with those in Fig. 2. In Fig. 7 and 9 (¢eTc/¢) is assumed to be equal to 1 when T/r is 0. The general good agreement between theoretical values and experimental results justifies the use of (4) and (7) for the case when ae = 0 and also for the case when ae is + v e . In Fig. 7, 9 and 12, it can be seen that the shear angle decreases with the ratio T/r and t/T. This reduction of the shear angle plus the increasing influence of the edge force explain the increase of specific cutting energy when T/r

300

W.K.

LUK

1"4 EXPERIMENTAL RESULTS THEORETI CAL VALUES

1"3 ~eTC 1'2 0

0 °

04 0

I

I

02

0"4

t

I

0.6

0"8

T r FIG. 7. a t = 0, r = 21-4 × 10 a in., full d e p t h cuts.

2'4. o o EXPERIMENTAL RESULTS THEORETICAL VALUES

2.0 k'r k

0o

1"6 0

1"2-

(2)

0"80.4 0

[

I

I

1

0.2

0.4

06

0.8

T r FIG. 8. ~Xe = 0, r = 21"4 × 10 - a in., full d e p t h cuts.

The Mechanics of Symmetrical Circular Form Tool Cutting

1.5

e o EXPERIMENTALRESULTS THEORETICAL VALUES

301

1

/ I

1"2

'/'eTC ,/,

i.i - ~ ' ~ , o. ' o-

I

o

o

o.

0"9

0-8

I 0"2

0

I 0"4

I 0"6

I 0"8

T f

FIG. 9. ~e = 22"1 °, r = 40 x 10 -a in., full d e p t h cuts.

1"6 1"4-

o m

o o EXPERIMENTAL RESULTS THEORETICAL VALUES

RT

k

1"2 I

o

0"8-

0"60

I

0.2

I

I

0-4

0.6

1

0.8

T r FIG. 10. ae = 22"1 °, r = 40 × 10 -a in., full d e p t h cuts.

302

W.K.

1.6 kr k

o o

LUK

EXPERIMENTAL RESULTS THEORETICAL VALUES

1"40

1"21_- o

o~

0'8!

I

1

0"6

0"2

0.4

I ~ ~ T

0'6

08

I

r FIG. 11. ~e = 22"1 °, r = 62 X 10 -8 in., full d e p t h c u t s .

1.5

o o EXPERIMENTAL RESULTS

1"4~etc i-3-

1'2I-II0.9

0

I

I

r

I

0"2 0.4 t 0.6 08 T

FIG. 12. ae = 22"1 °, r = 6 2 X 10 - a in., n o n - f u l l d e p t h c u t s .

The Mechanics of Symmetrical Circular Form Tool Cutting

303

and t/Tdecrease (Fig. 8, 10, 11 and 13). In full depth cuts, when T/r decreases, it is equivalent to a symmetrical vee form tool with the included angle 28 increasing. In non-full depth cuts, a symmetrical circular form tool is equivalent to a symmetrical vee form tool. The results shown here are consistent with those of symmetrical vee form tool cutting [1]. e e EXPERIMENTAL ]'5

-

--

THEORETICAL

RESULTS VALUES

kt

k

I-5I'10.9 07 0

I 02

I 0.4

i 0-6

i 0.8

t T FIG. 13. ~e = 22"1°, r = 62 × 10-3 in., non-full depth cuts. CONCLUSIONS 1. The analysis reveals the close relationship between the mechanics of orthogonal cutting and that of symmetrical circular form tool cutting. Once the data in orthogonal cutting has been collected, the magnitudes of the corresponding quantities in symmetrical circular form tool cutting may be predicted in straightforward manner. 2. The shear angle decreases with the ratio T/r and tiT. This reduction of the shear angle plus the increasing influence of the edge force explain the increase of specific cutting energy when T/r and t / T decrease. 3. There is a great similarity between symmetrical circular form tool cutting and symmetrical vee form tool cutting. Their results are consistent with each other. REFERENCES [1] [2] [3] [4] [5]

W. K. LUK, Int. J. Mach. Tool. Des. Res. 9, 17 (1969). F. EUGENE,Proc. Int. Prod. Engng Res. Conf., 1963, pp. 72-75, ASME (1963). E. H. LEE and B. W. SHAEFER,J. appl. Mech. 73, 405 (1951). E. J. A. ARMAREGO,Int. J. Mach. Tool Des. Res. 7, 23-37 (1967). W. K. LUK and R. F. SCRLrrTON,Int. J. Prod. Res. 6, 197 (1968).