Metal cutting analyses for turning operations

Int. J. Mach. Tool Des. Res.

Vol. 10, pp. 361-379.

Pergamon Press 1970.

Printed in Great Britain

METAL CUTTING ANALYSES FOR T U R N I N G OPERATIONS E. J. A. ARMAREGO* (Received 10 December 1969) Abstract--Two thin shear zone analyses are developed for turning operations. The anlyses cannot predict the deformation and cutting forces without experimentally determining some cutting parameters. No theoretical justification for the chip flow rule ~c = i has been found. A numerical investigation has shown that the power force in turning is mainly influenced by the chip length ratio, irrespective of the analysis used. Methods of applying these analyses for predicting the forces in turning are discussed. NOMENCLATURE

V~ V~ V~

1/I (X

8a, 82 8m O~m

C~ C~

f d T A

~e '7"

p Ss~ Ss2

s~l sL o ' n l , o'n2

i

Work velocity Chip velocity Shear velocity Feed velocity Rake angle for triangular tool Triangular tool profile angles measured from the rake plane Maximum rake plan angle or workpiece inclination from the horizontal for triangular cuts British Maximum rake angle Normal rake angle to the side cutting edge Rake angle in the velocity plane Side cutting edge angle End cutting edge angle Feed per revolution in turning Depth of cut in turning Maximum groove depth for triangular cuts Area of cut Angles representing the cut proportions Shear angle in the velocity plane "Friction" angle in the velocity plane Maximum shear stress Hydrostatic stress Shear stress along shear velocity in "shear planes" 1 and 2 respectively Shear stress perpendicular to shear velocity in "shear planes" 1 and 2 respectively Normal stresses on "shear planes" 1 and 2 respectively Angle of obliquity of the side cutting edge Chip flow direction in a plane normal to Vw

* Department of Mechanical Engineering, University of Melbourne, Victoria, Australia. 361

362

~7c qll ~22 G33

l

E.J.A. ARMAREGO Chip flow direction in the rake face Chip flow direction--the angle between Vc and the normal to the side cutting edge measured in the rake face Principal stresses Workpiece length cut Chip length corresponding to l

rl

Chip length ratio - l c

As~, As, A~, A~, A1A2 ~F Fp

Areas of "shear planes" 1 and 2 Projection of "shear planes" 1 and 2 in the maximum shear stress plane Projection of "shear planes" 1 and 2 in velocity plane Friction force direction in rake face Power force colinear with Vw Thrust force in the velocity plane and normal to Fe Resultant side force perpendicular to the velocity plane Shear angle relation constants Resultant force Component of resultant force in the rake face (friction force)

F~ FR C1, C2 R F

INTRODUCTION STUDIES OF the mechanics of cutting are mainly based on the simple orthogonal cutting process. Thin shear zone and thick shear zone models have been considered for a number of years [1-7]. The thin zone models are favoured since these are often thought to be representative of the deformation at the usual cutting speeds. While these analyses cannot quantitatively predict the deformation and cutting forces without resorting to experimental determinations of one or more cutting parameters, they do explain the commonly observed trends. In recent years some attention has been given to the mechanics of double-edged tools [8, 9]. The thin shear zone models developed for triangular profiled tools performing full-depth (vee) cuts and non-full depth (chevron) cuts are very similar to those for conventional orthogonal cutting. These analyses, like all cutting analyses, are incomplete since the shear angle cannot be theoretically predicted. This is not unexpected when one considers the innumerable unsuccessful attempts made by research workers for over fifty years. The double-edge tool analyses are developed to the present state of knowledge of the mechanics of cutting and can be correlated to orthogonal cutting data and used to predict trends. In this paper it is proposed to develop thin zone analyses for turning operations and study the effect of the cutting variables on the chip flow direction and cutting forces. CUTTING MECHANICS FOR TURNING OPERATIONS The analyses for triangular tools [8, 9] can be modified for turning operations. The triangular tools used in the previous work are similar to lathe tools since both types consist of a common rake face, two clearance faces and two cutting edges. By altering the geometry of the cut taken it is possible to show the similarity between the triangular tools and the lathe cuts. Figure 1 shows a triangular tool performing a lathe type cut and a turning operation. Ignoring the tool nose radius and assuming the feed velocity VI in turning is small compared to the tangential velocity Vw it is seen (Fig. 1) that T corresponds to the

Metal Cutting Analyses for Turning Operations

363

0rn

0m T

82 Vw o

(a)

A~ / O~ d

Ce

Vf =

0

Vw

(b) Fie. 1. Similarity between triangular tool and a lathe tool. (a) Triangular tool performing a lathe type cut, (b) turning with a British Maximum rake specification tool. d e p t h o f cut d a n d A D is equivalent to the feed p e r r e v o l u t i o n f . I n a d d i t i o n the r a k e angle a is equivalent to the British M a x i m u m r a k e angle a,, [10] while 31 = C8 - - Or,

(1)

3z = 90 - - (Ce - - Ore)

(2)

where Om = w o r k p i e c e inclination angle = m a x i m u m r a k e p l a n angle Cs = p l a n a p p r o a c h o r side cutting edge angle Ce = p l a n trail o r end cutting edge angle. T h e t r i a n g u l a r t o o l n o m e n c l a t u r e a n d the British M a x i m u m r a k e specification can be related to the n o r m a l r a k e angle specification, i.e. t a n i = t a n a cos 31 = t a n am cos (Cs

--

(3)

Ore)

and tan a sin 31 t a n an = V'I + t a n z a cos ~ 31

t a n am sin (Cs - - Ore) ~/1 + tan 2 am cos 2 (Cs - -

where i = angle o f obliquity o f the side cutting edge an = n o r m a l r a k e angle o f the side cutting edge

Ore)

(4)

364

E.J.A. ~ e . r 6 o

Having established the relationships between the various specifications, it is now possible to consider the deformation in turning operations based on the triangular tool geometry. This geometry leads to simpler analytical expressions.

Thin plane deformation models As in the previous work [8, 9] it is assumed that the deformation in turning can be represented by two "shear planes" pivoted about the two cutting edges. It is also assumed that a single chip is formed and that the chip velocity is fixed in magnitude and direction throughout the cut, (i.e. the deformation occurs on a series of parallel planes formed by the chip, work and shear velocities). Figure 2(a) shows the tool rake face and the shear planes from a number of views, together with the stresses on elements close to the shear planes. The chip flow direction is chosen in some direction f~ to the vertical plane. Area ADCBO represents the area of cut, AD is equivalent t o f a n d the projection of OA on a line perpendicular to A D is the depth of cut d. The two "shear planes" A D C O and BCO meet at OC. The angle between OC and the work velocity Vw measured in the velocity plane through OC is the shear angle Ce. This shear angle is constant at any other point on the cutting edges since Vc and Vw are fixed in magnitude and direction hence the shear velocity is also fixed (or from descriptive geometry the intersection of a (shear) plane by a series of parallel (velocity) planes results in a number of parallel lines). From velocity considerations it follows that

Vc vw

or

sin Ce cos ( 4 , -

tan Ce -1

lc ~,)

l

(rt) cos ~ -rt sin ~e

(5)

where Ce = shear angle in the velocity plane ~e = rake angle in the velocity plane

lc = chip length l = work piece length cut corresponding to lc

lc

rt = chip length ratio -- l

from the geometry of figure 2a it can be shown that tan ~ ---- tan ~ cos f~

(6)

hence from (5) and (6) rl

tan

~e =

X/1 q- tan2 a cosZ f~ _ rt tan a cos f~

(7)

Equation (7) shows that Ce can only be found if rt and f2 are known, since ~ is the given rake angle. Thus the deformation depends on the shear angle and the chip flow direction. Applying plane strain conditions (dcz2 = 0) as in the previous work [8] it can be shown that the stresses on the elements close to the "shear planes" are as given in figure 2a. From equilibrium of elements such as xl, x2 sectioned by the two "shear planes" it can also be

Metal Cutting Analyses for Turning Operations

365

shown that ~,~ = ~n~ = p

(8)

S ~ = S ~ ----0

(9)

and

where an1,

an

2 ~

s s , = ¢A¢ 1 [ A,s,1

(I 0)

Ss~ = "rAT2 [ ds~

(11)

normal stresses on the "shear planes"

S~.~, S ~ -----shear stresses in the "shear planes" perpendicular to the shear velocity Vs

Ssl, Ss~ = shear stress in the "shear planes" along the shear velocity Vs ~- = maximum shear stress

As~, As2 = areas of "shear planes" ATe, A,2 ----projections of shear plane areas in the maximum shear stress plane p -- hydrostatic stress = (all + cr33)/2 = ~zz. The stress distributions on the "shear planes" are uniform.

Merchant type solution case I For the Merchant type solution it is assumed that the resultant force acting on the chip at the "shear planes" is balanced by an equal, opposite and colinear force at the tool-chip interface. It is also assumed that the cutting edges are sharp and no edge forces are present. If it is considered that the resultant force acting on the chip at the "shear planes" is parallel to the velocity planes then the side forces Frl and Fr~ perpendicular to the velocity planes must cancel each other. This requires that the projected areas of the shear planes in the velocity planes be equal since equal hydrostatic stresses act on these areas. For the configuration shown in Fig. 2a the areas AiOiCiD2 and 02CiBz must be equal. The general conditions lead to various solutions depending on the proportions of the cut. For small feed to depth ratios (as is common in turning practice) giving small or negative values of fi and large 0 the chip flow angle £) will be greater than fl as in Fig. 2a. Point C will lie somewhere between points D and B (Fig. 2a). For large f/d ratios (large positive fi and small 0) ~) will be less than /3 as in Fig. 2b. The limiting case occurs when ~ ~-- fi as in Fig. 2c. The proportions of the cut are represented by/3 or 0 where tan fi ~- (f/d) cos Omcos (31 + Ore) -- sin 31 cos 31 + (f/d) sin Om cos (31 + Ore) tan 0 = [sin (31 + 32) cos 31 -- (f/d) cos 32 cos 2 (31 + Om)] [(f/d) sin 3z cos ~ (31 + Ore) + sin (31 + 3z) sin 31]

(12) (13)

and fi and 0 are related by sin (31 + 3~) cos (fi -- 0m) cos (3z -- 0) -sin (31 + fl) cos (31 + Ore) cos (31 + 0)"

(14)

366

E.J.A. ARMAREGO

Ai

Dl Ci

Bt

Vw 0

Oi

0"53

,tT/B3 r ~ c

2

Lp°S20"n2

03C3 o-,~

0z~

P

Vw

End-on view of 'shear planes'

~

O.i

View in velocity plane "through 0

(a)

0

A

e

~

O,

02~ . ~ View in velocity plane fhrough 0 (b) Fla. 2. Thin plane deformation in turning for various cut proportions. (a) f/d small, ~ > ft. (b) f/d large, ~ < ft. (c) Limiting case, Q = /~.

367

Metal Cutting Analyses for Turning Operations

Ore,

L--- ~°a

\

Ci Di

,

/A_/

~

v~

o

A ~

o~.

C D 22 v~

~Vw View in velocity plone through 0

FIG. 2 (C) For small feed to depth ratio f~ > fi and the projected areas A2D2C202 and B2C202 in Fig. 2a will be equal when cos (82 -- f~) + tan a tan 4'e cos 32

= [2 cos (82 -- 0) sin (31 3- 82) cos (f2 -- Om)t ~ cos (81 + 0) -- cos (81 + Ore)sin (31 + f2)J × {cos (81 3- Y~) + tan a tan Ce cos 31}

(15)

or

cos (82 - ~) + tan ~tan ¢ eCOS8~ = [ 2 y o s ( ~ -- 0~) cos ( ~ -- 0~) 1 si~(8 ! ± 82) (sin(313-fl) -- s i n ( 3 1 3 - ~) j cos (813- 0m) × {cos (81 3- £~) 3- tan c~tan Ce cos 31}.

(15a)

For large feed to depth ratios f) < fl and the projected areas A202C2, B202C2D2 in Fig. 2b will be equal when [s!n (fl 3- 81) sin (3~ -- fl) cos (fl -- Ore) sin (fl -- f2)] sin (3~ + 8 2 ) . . . . -tcos ( ~ ~ 0m) ] × [tan ¢~ tan ~ cos 3z + cos (Sz -- ~)] cos 2 (fl -- Ore) sin (Sz -- ~) = cos (~) ~ 0m) cos (3~ 4 0 m ) [tan ¢, tan ~ cos 81 + cos (8~ + ~)].

(16)

F r o m equations (15) and (16) it is seen that ~ will depend on the chip length ratio rt (or Ce) and the given tool and cut geometry. Thus the chip flow direction Q can be found if rt is known and vice versa. Although it is most likely that equation 15 (or 15a) will give the necessary solution for for the u s u a l f / d ratios in turning, in general some method is necessary to indicate whether equations (15) or (16) are applicable. Considering the limiting case when ~ = / 3 as seen in

368

E.J.A. ARMAREGO

Fig. 2c it may be shown that areas A20~C2D2 and B2OzCzD2 are equal when tan dee tan ~ sin 0 = sin (t) -- 0)

(17)

or substituting for dee from equation (7) tan e sin

[

r~

0 X / 1 - - t a n 2 a c o s ~ f~ - - rz t a n ~ c o s f~

] = sin(f~ -- 0).

(17a)

Equation (17) can also be found from equations (15a) and (16) by substituting/3 = f~ and using equation (14). For any given chip length ratio rt and cut geometry (i.e. 0 from equation (13)) the chip flow direction f2 can be found from equation (17a). This value of f~ corresponds to the limiting value of/3, say ilL. If /3z is greater than /3 found from equation (12) then equations (15) or (15a) apply. If/3L < /3 actual then equation (16) applies. An alternative approach is to substitute /3 = f~ in equation (17a) and determine the limiting value of rt, say rlL. If rIL < actual rt equations (15) or (15a) apply. The forces acting on the tool for an assumed chip flow direction f2 are shown in Fig. 3. The resultant force R is parallel to the velocity planes and made to pass through the tool point O for convenience. The force components along and perpendicular to the work velocity in the velocity planes are given by Fp =

.

(18)

~'A cos(Ae - - ae)

Sln deeCOS(dee -I- ,~e -- ~e) FQ =

. "cA sin (Ae - - ~e) sin deeCOS(dee -~- ,~e -- ~Xe)

(19)

where A = Area of cut Ae ---- "friction" angle in the velocity plane. The area of cut A is found from the cut geometry so that A = df-

f 2 cos (81 +

Ore) COS

(32 --

Ore)

(20)

2 sin (81 + 82) From the above equations it can be seen that the chip flow direction f2 can be found when the chip length ratio rz is known. The forces can only be found when an extra unknown (i.e. he) is given. Assuming a shear angle relation of the type dee = C1 - - Cz(ae - - ae)

(21)

where C1 and C2 are known (or experimentally found) constants, the chip flow direction and the forces can be determined when rt is given. It is recognized that the above analysis is a simplified model. In fact this process is a complex three dimensional plastic deformation problem. Two points, however, are worth noting; the side forces Frl and Fr~ are equal and opposite but not colinear--thus the resulting couple has been ignored. In addition, the resultant force component along the rake face F (friction force) is not generally colinear with the chip velocity. From Fig. 3 it can be shown that FQ s i n f~ _ tan (he - - ~e) s i n f~ tan ~ v = F p sin a + FQ cos f~ cos ~ -- sin a + tan (),e -- ~e) cos f~ cos

(22)

Metal Cutting Analyses for Turning Operations

369

View X (true view of rake plane)

/ B3

0

~

// lle

Iq

Ol

~Vw

View in velocity plone "l'hroucjh 0 (resul't'an'l" force R in velocity plane)

F~o. 3. Forces and chip flow directions in the rake face (case I).

and tan f2c = tan f~ cos ~

(23)

where f2F --~ direction o f friction force in the rake face f2c -----chip flow direction in the rake face. Equating equations (22) and (23) so that f2F = ~qe and using equation (6) it follows that ,ke = 90 °, i.e. the resultant force R is colinear with the chip velocity Vc. It can also be seen f r o m equations (22) (23) and (6) that f~e I-- f~c if ~ = O or f~ ~-- O. The condition ke = 90 ° is unrealistic. When/3 = 32 and Om = O the lathe cut becomes a full depth asymmetrical triangular cut and f~ ---- O is a reasonable solution. This condition also makes the side forces Frl and Fr, equal, opposite and colinear. The conditions ~ = O and f2 ---- O are therefore special, rather than general, solutions for this analysis. In general it is not possible to eliminate the resulting couple caused by the side forces although the analysis can be modified so that f~F ---- f2e. The conventional force c o m p o n e n t s for turning can be found by resolving the forces Fp and Fo--i.e. F t a n g e n t i a l ~--- Fp

Freed

=

Fradial =

FQ sin (f2 -- Ore) F 0 COS ( ~

--

(24)

Om).

Merchant type solution case H In this model the assumptions for case I apply except that the friction force and chip velocity are also assumed colinear. F o r f~c = f~e the resultant force R is not parallel to the 24

370

E . J . A . ARMAREGO

velocity planes and a side force FR = F q - - Fr~ exists. The force equations become __ r A c o s (me - - ~e) sin ~e c o s me

(25)

rA sin (Ae -- O~e) r A sin (~Oe -- Ce) FO = sin Ce cos (¢e + Ae -- CXe)~ sin Ce cos O~e

(26)

FR = Fr~ - - Fr~ = p[A1 - - A2]

(27)

~Oe=Sbe+)~e--~e=tan-l(p)

(28)

"cA c o s ()~e - - ae)

F p = sin ~e c o s (t~e -~- Ae - - O~e) - -

where

A1, Ag = projected areas of shear planes 1 and 2 on the velocity planes F t t = side force component perpendicular to the velocity planes. The friction force direction in the rake face, f~e, is given by tan f2F =

Fo sin f2 + FR cos f2 (F 0 cos f~ -- FR sin f2) cos a + F p sin a

(29) (cf 22)

while the chip flow direction in the rake face f2e is found from equation (23). The appropriate projected areas A1 and A2 should be used depending on the cut proportions. For f2 > fl (Fig. 2a) A1 = Area A2OgCzDg _ O D 2 12 cos (/3 - - Ore)

2

cos (£2 ~ Ore)t [ cos (81 + ~) sin g (31 +/3)t sin (31 + ~) J tsin (311-+ ~ (8~ 7+ Om)J

~ sin(31 +/3)

{

tancos31}

(30)

× cot Ce + cos(81 + ~)

oo2/sio2 ,l+ cos(,2- ~2)}{cot + tanocos, , (sin (81 + 82) sin

A2 = Area B202C2 = - ~ -

(31 ~-

COS (32 - - ~'~2)J

for f~ < / 3 (Fig. 2b) 0m) cos (31 + 0ram) c°t Ce + ~

Al=A2OgCg=-2-'cos(~

(81+~-)

O D z tsin (81_+ fi) sin ($g --/3) cos (/3 -- Ore) sin (/3 -- ~)} A2 = B202CgD2 = ~ ~ sin (81 + 32) + cos (f2 -- Ore)

{

tan os, }

× cot ¢~ + cos ~z -~ ~-) cot (32 -- f2)

(33)

for f2 = / 3 (Fig. 2c) A1 = AzOgCgDa = ~ A2 = B202C2D2 =

OD2

2

i( os o-0o, os + o,} { Ot e + cos tan(31cos l} ~ s (81-T - O ~ + f*j {s i n ( a ~ +sinf ~(31) c o+s ( a32)g _ ~) }{c o t ¢ ~ + c ota. s ( 3 zcos,,~ )}

,,,,

Metal Cutting Analyses for Turning Operations

371

The area of cut A should also be expressed in terms of length OD for use in equations (25-27), thus

A = ODZ sin (31 ÷ / 3 ) sin (32 -- /3) ÷ cos (fi --_ Ore) sin (81 + /3)I --2 .....

sin (31 -~- 32)

COS (31 ~-i

Ore)

(36)

)"

In order to show the condition f~F = f~c it is necessary to combine the appropriate above equations. Due to the complex equations for A, A1 and A2 required to express FR and hence ~F, it is more convenient not to combine the equations into a single relation. The various substitutions can best be done numerically when solving a particular problem. It will be noticed that f~F is a function of fl, rt (or ~e) and Ae (or We), while f2c is a function of fl, (other variables such as ~, 31, 32 being known). Thus the chip flow direction f2 can be found if two of the three unknowns are given or experimentally found. This is in contrast to Case I where rt (or Ce) was sufficient to find f~. The extra unknown he (or OJe) is required because ~)F and hence f~ cannot be found unless the force components or their ratios are known. When a shear angle relation as in equation (21) is assumed, f2 can be found if rz (or Ce) is known. In solving the above equations it is necessary to choose the correct values of the projected areas A1 and Az. For smallf/d ratio it is expected that f2 > / 3 and equations (30) and (31) are used. In general, as in the previous case, the limiting condition f~ = /3 can be studied to decide which equations should be used for the projected areas. The conventional force components in turning become Ftang = Fe

Ffeea

]

FQ sin (f2 -- 0m) + FR cos (~2 -- Om)i

Fraaial = FQ COS (f2 -- Om) -- Fn sin (f~

(37)

Om)]

The apparent coefficient of friction/~ can be found in the usual way from the force components, chip flow direction and rake angle. The chip flow direction can also be defined as per oblique cutting with a single cutting edge (i.e. by Vc as shown in Fig. 3). Thus ;Te = 90 -- (f2c + 31')

(38)

31 = tan -1 (tan ~1 cos ~).

(39)

where The above analyses are thus developed to the same level as orthogonal cutting with a single cutting edge so that the forces and deformation cannot be determined without some experimental work. NUMERICAL ANALYSIS AND DISCUSSION Although the above analyses do not give complete solutions it is possible to numerically study the trends in chip flow direction f2 and ~c as well as the cutting forces. Computer programmes were prepared assuming two shear angle relations. By selecting a chip length ratio rt it was possible to study the effect of tool plane angles, rake angle, feed to depth ratios etc. The forces can be expressed in a dimensionless form, e.g. FP/rA, for comparison purposes. A sample of these calculated values are shown in Table 1. F r o m Table I it is noticed that for Case I the chip flow direction f~ (or ~7c)and shear angle Ce are independent of the shear angle relation constants used. For Case 1[ the constants C1 and C2 will affect ~, ~c and Ce.

9 10 11 12 13 14 15 16 17 18 19 20 21

8

2 3 4 5 6 7

1

Run

4,

2O

10

60 45 3O 10 5 60 45 3O 10 5

60

60

10

10

81

~

5'04 7"11 8"68 9"85 9"96 10"31 14"43 17"50 19'72 19'93

•04

5"04

i

4, 0-05

4, 0"05

I --59-27 --43"53 - - 2 7 " 80 --7.20 --2"15 --59.27 --43" 53 --27"80 --7"20 --2"15 4'

0"3

0'3 0"4 0'5 0"6 0'3

17"25 17"25 17-26 17"26 17'27 17.30 17"31 17"25 22'81 28"10 33"05 17"25 17"17 17"05 16"82 16"75 17"46 17"42 17"27 16"85 16"71

0'3

--59"27 58' 50 --56"87 --53" 10 --42"99 --7.52 15"00 --59"27

0-05 0"10 0-20 0"40 O' 80 1" 60 2"00 0"05 --

$e

r~

fl

f/d

~le --0'42 --0"02 +0-74 +2'45 +6"08 +15'50 +22"01 --0'42 --0'92 --1 '51 --2-11 --0"42 --0'50 --0"81 --1"3 --1"36 +0"38 --0"22 --1"52 --3"85 --4'48

~ 31'19 30"79 30-02 28"30 24"64 15"10 8"50 31'19 31 "69 32"29 32'89 31" 19 46"38 61"55 81"57 86'49 32"79 48"78 64"45 84"77 89"79

CASE ] - - R E S U L T A N T FORCE IN THE VELOCITY PLANE

6.44 6"44 6"44 6"44 6"43 6-42 6"42 6"44 4"76 3"75 3"07 6"44 6"47 6"52 6"62 6"65 6"36 6"38 6"43 6"60 6'66

4"22 4"22 4"22 4"22 4"22 4.21 4"21 4"22 3"38 2"87 2"54 4"22 4"24 4"26 4"31 4'32 4"18 4"19 4"22 4"30 4"33

Fp/~'A~f Fp/~'A*

TABLE 1. (BASED ON TRIANGULAR TOOL GEOMETRY.) NUMERICAL EFFECT OF TOOL AND CUT GEOMETRY AND CHIP LENGTH RATIO ON THE SHEAR ANGLE, CHIP FLOW DIRECTION AND POWER FORCE IN TURNING.

o

Im

--4 t,~

60

60 45 30 10 5 60

10

10

+ 5.04 7.11 8-68 9.85 9.96 10.31 14.43 17.50 19.72 19.93

I

5-04

5" 04

i

4, 0"05

0'05

--59.27 --43.53 --27.80 --7.20 --2"15 --59"27 --43-53 --27.80 --7.20 --2'15

0" 3

--59"27 --58"50 --56'87 --53"10 --42.99 --7'52 15"00 --59"27

0-05 0-10 0.20 0-40 O. 80 1.60 2.00 0.05

f~t 28'84 28-49 27"81 26"20 22"82 13'90 7.70 28"84 28.34 27"84 27'34 28.84 42.99 57"13 75-99 80'75 29.64 43"59 56"93 73"13 78"75

~et 17'26 17'26 17.26 17"27 17.28 17.30 17"31 17.26 22"84 18"16 33.16 17.26 17-19 17'09 16'89 16"83 17"46 17.44 17-35 17-12 17'00

1.91 2'25 2'93 4-53 7"88 16-69 22'80 1-91 2'40 2"90 3"39 1"91 2"89 3"65 4'37 4"47 3"43 4.97 6.24 8"48 7.26

~et 6-44 6.44 6.44 6"43 6'43 6.42 6.42 6'44 4'75 3.74 3'06 6-44 6"46 6-51 6'59 6-61 6.36 6"38 6.40 6"49 6'54

Fv/~At 17"27 17"27 17.27 17"28 17"29 17"30 17'31 17.27 22"85 28.18 33"19 17.27 17.22 17"13 16.96 16-91 17.46 17-45 17'42 17.28 17'22

(ae* 25"94 25"59 25'01 23'60 20.52 12'40 6"80 25'94 25"94 25'94 25"94 25.94 38"89 51.93 69'69 74'35 24.44 36'29 47.93 63'59 67'65

~* 4'78 5-13 5.71 7.10 10'15 18"16 23.68 4'78 4-78 4'78 4-78 4'78 6'98 8.88 10'75 10.96 8"44 12.18 15-37 18.45 18.94

~e*

t Shear angle relation constants C1 = 45 °, 6"2 = 1/2 (Ore = 0"0, 32 = 60°).

4, 0"3

0-3 0"4 0"5 0'6 0-3

r~

fl

f/d

* Shear angle relation constants C1 = 45 °, Cs = 1.

30 10 5

60

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

20

31

~

Run

CASE I I - - f ~ = f~v

4'22 4.22 4'22 4"22 4.21 4'21 4"21 4'22 3.73 2'87 2-53 4"22 4-23 4"24 4"28 4'29 4-18 4.18 4.19 4'22 4"23

Fp/zA*

ta,a

0

©

~q e-

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E.J.A. ARMAREGO

From this numerical analysis a number of interesting observations can be made. Considering runs 1-7 it is seen that by increasing the feed to depth ratio from 0-05 to 2.0 very small changes in Ce and Fpfi-A occur when the chip ratio r~ is held constant. Large changes in f2 and ~¢ occur and as expected f~ gets smaller as f / d increases since the cut tends to an asymmetrical full depth triangular cut. The same trends have been shown to occur for different levels of rt, c~and 31 (or Cs). Large changes in 31 (31 = Cs when Om -- O) cause little change in ~e, Ce and Fp/.rA when the chip length ratio is constant. In varying 31 from 60 ° to 5 ° Fpfi-A increases by up to about 3 per cent depending on the shear angle relation constants used. The chip flow direction given by fl increases as 31 decreases. Again this is expected since whenf/d is small and the chip flow direction is roughly perpendicular to the major cutting edge. Hence ~e is small and 31 + f~ is close to 90 °, i.e. f2 increases as 31 decreases. These trends can be seen from runs 12-16 for ~ = 10 ° and again from runs 17-21 for ~ = 20 °. In changing the rake angle ~ from I0 ° to 20 ° there is a slight increase in Ce and a small decrease in Fp/TA when rt is constant. These changes are of the order of a few per cent. Relatively little numerical changes in f~ and ~e are shown. These results can be seen by comparing corresponding runs such as 12 and 17 or 16 and 21. From runs 8 to 11 it is seen that changing the chip length ratio from 0.3 to 0.6 increases the shear angle Ce and the force parameter Fp/-cA decreases considerably. Little changes in f~ and ~c occur. From an overall look at Table 1 other important conclusions can be made. (i) The shear angle Ce and the force parameter Fp/'rA are virtually independent of the analysis used, other variables being constant. (ii) The chip flow direction as given by f~ and ~c depends on the solution used and the shear angle relation constants. (iii) The shear angle Ce is practically independent of f~ and ~/c and is greatly affected by the chip length ratio. (iv) For a smallf/d ratio of 0.05 Stabler's chip flow " L a w " (i.e. ~e = i) does not apply in general. The case II solution together with shear angle relation constants C1 = 45 ° and C2 = 1 approximates the chip flow "Law." (v) The force parameter Fp/-rA is almost independent of the rake angle ~, the plan angle 31 (or Cs) and the feed to depth ratio but is dependent on the chip length ratio rt. The shear angle relation constants will influence this force. Thus the force parameter Fpfi-A may be expressed as

Fp/~-A = function (rz).

(38)

Equation (38) is shown in Fig. 4 for a variety of rake angles, tool plan angles, and feed to depth ratios. It must be realized that the rake angle variation will most likely affect the deformation and the chip length ratio rz so that Fp/'rA will vary with ~ experimentally. Similar effects can occur when the other variables are altered but it is important to recognize that the numerical changes in these variables do not in themselves greatly alter Fp/-rA. The chip length ratio r~ is thus shown to be of fundamental importance. The results in Table 1 are based on the triangular tool nomenclature. Since this specification can be related to the British Maximum rake and the normal rake specification it is possible to consider these two lathe tool specifications in a similar way. Table 2 in Appendix 1 shows the numerical results for a small f / d ratio, from which the main trends discussed above are shown to apply for any specification. Table 2 also shows the feed and radial forces in a

375

Metal Cutting Analyses for Turning Operations 7

Cj = 4 - 5 " ,

C2= I / 2

,L C 1=45., Cz=l

0 0,2

I

0.3

I 0,4. Chip

l 0.5

length r o t i o

I 0-6

I 0-7

q

FIG. 4. Effect of chip length ratio on power force parameter for a range of conditions. (~, 0-20 °; 3t 60-5°;f/dO'05-2"O; Om0-15 °, bz = 60 °. Both cases included for each set of C1 and C2.)

dimensionless form. For any given chip length ratio and shear angle relation constants the two solutions give very similar numerical results. The feed forces Ffeed/'rA are slightly higher for Case II than Case I while the radial forces Frad/rA show the reverse trend. Since the above analyses cannot predict the shear angle, chip flow direction and cutting forces without resorting to experimental work, it is impossible to theoretically prove which lathe tool specification is the most relevant one from the mechanics of cutting point of view. Previous attempts at arriving at a lathe tool specification have been primarily based on the ease of grinding and inspecting these tools [11, 12]. Stabler [13, 14] claimed that the normal rake specification fulfilled another important condition--i.e, this specification was directly related to the mechanics of cutting. A study of the various lathe tool specifications has shown that a modified normal rake specification was the easiest to grind correctly since all the specified angles could be directly set on the vice axes. All other specifications required one or more corrections to arrive at the specified angles [15]. It may also be argued that all the specifications are relatively easy to grind correctly provided the geometry is fully understood. The necessary corrections can be made at the design stage followed by clearly set out methods of grinding and inspecting the tools. The normal rake angle specification was developed from a comparison with oblique cutting with a single cutting edge. The comparison may be valid whenf/d is small so that the secondary cutting edge effect is ignored. Theoretically the mechanics of oblique cutting cannot predict the chip flow direction or the cutting forces. The fact that ~7cis approximately equal to i (and slightly affected by an) and that an is mainly responsible for controlling the

376

E.J.A. ARMAREGO

power force Fp is purely experimental (16, 17, 18, 19). Tests with lathe tools ground to the modified normal rake specification also show that an controls the power force irrespective of Cs and i for small f / d ratios (9). These fortuitous experimental trends are of practical importance since the power force depends on an and the chip flow direction is mainly controlled by i. Hence two tools with the same an will need the same power when the area of cut and cutting speed are the same. For largef/d ratios the oblique cutting trends do not necessarily apply since the second cutting edge can play an important role. The above analyses can thus be of use. From a theoretical point of view the rake angle, whichever way specified, is one of many variables which affect the tool geometry. Experiments with full-depth triangular tools have clearly shown that rz (and Ce) varies with ~ and 3. Non-full depth cuts give lower r~ (¢e) values than full-depth cuts for the same tool geometry. Generally there is a transition between fulldepth and non-full depth cuts with rt (and Ce) rapidly reaching a lower constant value as t/T decreases (8). These tests have shown that the usual relation between Ce and (Ae -- ~e) applies. The changes in rt and Ce are related to the changes in friction at the tool-chip interface, which varies with rake angle, tool profile, and cut proportions. From the numerical analysis it is expected that the variables which influence rl will also affect the power force and other force components. Whether or not it can be experimentally shown that one, and only one, rake angle controls the chip length ratio and the power force is subject to speculation. Nevertheless, these analyses help bridge the gap between theory and practice and provide guidelines for empirical and semi-empirical tests. Empirical tests can be run to determine the constants in equation (38) or some similar curve e.g. Fp/rA -- rz. Thus the power force for a given cut can be found from a simple chip length ratio measurement. An alternative and more fruitful approach is to run simple orthogonal cutting tests to determine basic cutting parameters such as shear stress and shear angle relation constants. The forces in turning can be found from either analyses discussed above by using the "standard" cutting data and the chip length ratio measured for the lathe cut in question. Slight corrections for tool edge forces will also have to be made as shown in [8]. The orthogonal cutting tests for obtaining the "standard" data may be likened to the materials tests used in stress analysis for obtaining the stress-strain curve and yield criterion to analyse more complex problems.

CONCLUSIONS (1) TWO thin shear zone analyses are developed for turning operations. (2) These analyses, like all simpler cutting analyses, cannot predict forces, chip flow directions or shear angles without resorting to experimental tests. (3) There is no theoretical evidence to suggest that ~c = i w h e n f / d ratios are small. (4) A numerical investigation has shown that the dimensionless power force Fp/rA is numerically dependent on the chip length ratio and independent of the tool and cut variables. Experimentally, variables which influence the chip length ratio will affect the power force. (5) The power force parameter Fp/rA is independent of the analysis used but depends on the shear angle relation constants. (6) Methods of using the developed analyses for predicting the forces components in turning are outlined. Both analyses will yield very similar results.

Metal Cutting Analyses for Turning Operations

377

REFERENCES [1] V. PIISPANEN,Teknill. Aikak. 27, 315 (1937). [2] M. E. MERCHANT,J. appl. Phys. 16, 267 and 318 (1945). [3] E. H. LEE and B. W. SHAEEER,J. appl. Mech. 18, 405 (1951). [4] W. B. PALMERand P. L. B. OXLEY,Proc. Instn mech. Engrs 173, 623 (1959). [5] K. OKUSHIMAand K. HITOMI,J. Engng Ind. 83, 545 (1961). [6] P. L. B. OXLEYand A. P. HATTON,Int. J. Mech. Sci. 3, 68, (1961). [7] S. KOBAYASHIand E. G. TrIOMS~N,J. Engng Ind. 81, 251 (1959). [8] E. J. A. ARMAREGO,Int. J. Mach. Tool Des. Res. 7, 23 (1967). [9] E. J. A. ARMAREGOand R. H. BROWN, The Machining of Metals, Prentice Hall (1969). [10] B. S. 1886 (1952). [1 l] D. F. GALLOWAY,"Standardization and Practical Application of Cutting Tool Nomenclature", Res. Dept. Inst. Prod. Engrs. [12] D. F. GALLOWAY,Proc. Instn mech. Engrs 168, 67 (1954). [13] G. V. STABLER,Proc. Instn mech. Engrs 165, 14 (1951). [14] G. V. STABLER,Proc. lnstn Prod. Engrs 34, 264 (1955). [15] E. J. A. ARMAREGO,Int. J. Mach. Tool Des. Res. 4, 189 (1965). [16] E. J. A. ARMAREGOand R. H. BROWN, Machine Shop andMetal-working, Australia (Oct. 1963). [17] R. H. BROWNand E. J. A. ARMAREGO,Int. J. Mach. Tool Des. Res. 4, 9 (1964). [18] J. K. RUSSELLand R. H. BROWN, Int. d. Mach. Tool Des. Res. 6, 129 (1966). [19] G. V. STABLER,Advances in Machine Tool Design and Research. Pergamon Press, Oxford (1964).

1 ~

c~m

10 10 10 10 10

5 10 15 20 25

10 10 10 10

Run

1 2 3 4 5

6 7 8 9 10

11 12 13 14

--20 --10 0 10

0 0 0 0 0

0 0 0 0 0

Om

30 30 30 30

30 30 30 30 30

45 30 15 10 5

Cs

r

17-20 17"14 17.05 16"94

16"89 17"05 17.17 17.27 17'34

17'17 17'05 16'88 16'82 16"75

~e

40'87 51'17 61"57 71'77

60'17 61'57 62'97 64"47 66-07

46"39 61"57 76"53 81"51 86.53

~2

--0"01 --0-31 --0'08 --1"23

0"02 --0-83 --1"31 --1"54 --1"54

-0"51 --0"83 --1.11 --1-23 --1-41

4"23 4.24 4'26 4-28

4'29 4-26 4'24 4"22 4"20

4'24 4.26 4"30 4"30 4'32

1"95 1"97 1"99 2-01

1"99 1-99 1"99 2-00 2'01

1"62 1'99 2-32 2"28 2.32

1'09 1-08 1"08 1-08

1'14 1"08 1'02 0"96 0"89

1"54 1-08 0"54 0'34 0.14

Fp/7"A Ffeed/'rA Fraa/TA

17'24 17-19 17'13 17'06

16'92 17-13 17"31 17'42 17.46

17'22 17"13 17"01 16"96 16.91

~e

34-74 43'24 51"94 60"74

54'94 51"94 49.54 47"94 46'94

38"89 51"94 65.16 69"71 74'37

~

6"11 7"63 8"87 9"92

5.26 8"87 12.30 15'36 18'26

6'97 8"87 10"37 10"73 10"94

~e

4-22 4"23 4"24 4'26

4'29 4.24 4"21 4.19 4-18

4-23 4'24 4.27 4'28 4"29

Fp/rA

A

"qc

Case II, f~c = f~F

X. aC

Case I

2.04 2'07 2-11 2'15

2'05 2"11 2'18 2'25 2-33

1.73 2.11 2'34 2.38 2'40

0-97 0"96 0"95 0"95

1.07 0'95 0-85 0"77 0"70

1"47 0'95 0-36 0"15 --0"07

Freed/'rA Frad/'rA

TABLE 2. NUMERICAL EFFECT OF TOOL GEOMETRY ON THE FORCES IN TURNING. Ce 15°,f/d = 0"05, rz = 0 " 3 , C1 = 45 °, C2 = 1. (a) BASED ON BRITISH MAXIMUMRAKE SPECIFICATION

APPENDIX

c)

.>

Do

5 10 20 25 10

10 10 10

6 7 8 9 10

11 12 13

10 20 25

10 10 10 10 5

30 30 30

30 30 30 30 30

45 30 15 10 5

10 10 10 10 10

1 2 3 4 5

10 10 10 10 10

Cs

Run

17'30 17.26 17-21

17.04 17'30 17'44 17.29 17'31

17.30 17"30 17"30 17"31 17'31

46"57 68.00 74.41

65'22 46-57 27'37 22.27 26.71

46"99 46'57 46-33 46"21 46'23

f~

--0-22 --1'92 --3"08

--1.12 -0'22 1"67 2.54 0"51

--0.64 --0.22 0-02 0'14 0"12

_sk

Case I

BASED

ne

(b) RAKE

4.21 4'22 4-23

4"26 4"21 4-18 4.21 4"21

4-21 4.21 4.21 4'21 4-21

1.95 2.02 2"95

2-00 1'95 1.93 1'95 1-92

1-63 1'95 2.15 2'19 2-21

1.04 0"93 0"87

1-06 1'04 1"03 1.04 1-09

1"50 1'04 0-51 0'32 0.13

Ce

17.36 17.44 17.46

17.15 17.36 17"42 17'26 17.32

17.36 17.36 17.36 17"36 17-36

SPECIFICATION

Fp/'rA F~eee,/'rA Fraa/'rA

ON NORMAL

37"24 49'50 52.01

54.06 37"24 21-09 17'14 22.91

37"19 37.24 37-46 37.61 37.67

~

9.08 17'20 20-75

10.15 9'08 7-63 7'23 4-27

9.12 9'08 8.85 8"71 8"65

"qc

4-20 4'18 4.18

4.24 4"20 4"19 4'22 4"21

4"20 4-20 4-20 4'20 4.20

2"08 2-31 2"44

2-14 2"08 2-05 2'07 2"00

1"78 2'08 2-25 2"27 2"27

0.90 0"73 0.65

0'92 0.90 0'88 0"89 0.97

1.41 0'90 0.31 0'11 0"09

FplTA Ffeed/TA Frad/VA

Case 1I, ~L = OF

5"

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