Dynamic cutting coefficients in three-dimensional cutting

Int. J. Mach. Tool Des. RBs. Vol. 20, pp. 235-249. Pergamon Press Ltd. 1980. Printed in Great Britain.

0020-7357/g0/1201-0235

$02.00/13

DYNAMIC CUTTING COEFFICIENTS IN THREE-DIMENSIONAL CUTTING V. GRASSO*, S. NOTO LA DEGA* and A. PASSANNANTI* (Received 9 July 1979; in final form 2 July 1980) Abatraet--A new theory to predict the dynamic cutting coetticients from steady state cutting data is developed. The derived expressions are able to take into account the effect of the main angles defining the tool geometry on the chatter stability limit and show a good agreement with experimental results. The proposed model allows the identification of a parameter as an index for chatter sensitivity of the work material.

NOMENCLATURE ao

F Fx F: F~ ho

h(t) ! T to

V Vc W Y Ye 6 8

),

Po p(t) ~m Ts

~e co

uncut chip section force along the rake face thrust force normal to the mean cutting direction tangential cutting force along the mean cutting direction force along ~ axis undeformed chip thickness dynamic undeformed chip thickness chip-tool contact length revolution time of the workpiece nominal feed/revolution cutting velocity vector chip flow velocity vector width of cut = coT- e, phase angle orthogonal rake angle normal rake angle rake angle in it plane slope of free surface phase angle chip flow angle in the rake face angle between ~ and X directions inclination angle of main cutting edge intersection axis between ~ and (X, Y) planes the plane definite by chip and cutting velocity vectors undeformed chip thickness measured in the 7c plane dynamic undeformed chip thickness variation in mean shear stress along the tool-chip contact area mean shear stress along the shear plane shear plane angle shear plane angle in 7t plane side cutting edge angle angular frequency of chatter

*Istituto di Tecnologie Meccaniche, FacolUi di Ingegneria, Viale delle Scienze, 90128 Palermo, Italy. 235

236

V. GRASSO, S. NOTO LA DIEGA and A. PASSANNANTI

INTRODUCTION CHATTER is one of the main factors which limits the efficiency of machine tools in metal cutting. Many studies have been carried out to predict the maximum unconditionally stable width of cut which a machine, characterized by its transfer function, is able to remove. The accurate prediction of this parameter depends on the precision with which the dynamic components of cutting force are known when cutting speed, feed and tool geometry are changed. The most widely accepted analytical expression for dynamic force, that was proposed by Tobias and Fishwick [1], takes into account both the stiffness and the damping of the cutting process during self-excited vibrations. Recently some models [2-4] for predicting the dynamic cutting coefficients from steady state cutting tests have been proposed, but these studies, in order to simplify the analysis, have been developed for orthogonal cutting. In particular the method developed in [3] takes into account the main factors that can influence the dynamic cutting force components; the used coefficients are determined from steady-state cutting tests except for the attenuation factor of force ratio dynamic variation, that must be obtained by means of dynamic tests. Moreover the model is not easily extensible to three-dimensional cutting as long as a reliable static model is not available to define an equivalent shear angle. The aim of this paper is to present a new analytical method for the prediction of the critical width of cut in three-dimensional cutting that does not require a precise determination of the shear angle. The analysis is based on the linear force model proposed for static cutting [5, 6] which permits to evaluate both the machinability of materials and the "sensitivity" to the cutting parameters. The expressions obtained for the dynamic force components can be used to compute the stability chart in turning for every combination of the tool angles and cutting parameters. The critical widths of cut thus predicted are in good agreement with experimental data.

STATIC THREE-DIMENSIONAL CUTTING It was shown that, for a wide range of speed and feed, in orthogonal cutting the inverse of the chip thickness ratio is linearly related to the parameter (F/Ao) ffi (~ml/to) which takes into account the variation both of the tool-chip contact length and of the mean shear stress Zmon the contact rake face [6]. Applying the minimum work hypothesis, it can be shown [5]* that the ratios of the cutting force components to the uncut chip area are correlated by linear relationships. Recently [7], the authors have shown that this linear model is also valid for threedimensional cutting, as can be seen in Fig. 1. Since both the value of residual specific force (Fz/Ao) ffi Mo for (F/Ao) ffi 0 and Ya ffi 0 and the slope B of the linear relationship between F,./Ao and F/Ao are independent of the tool geometry [7], these parameters appear as characteristic constants of the machined material. Particularly the value of Mo is a measure of the minimum work that must be employed to remove the material when the frictional force and the normal rake angle are zero. From such a point of view this parameter can be used as a machinability index of the materials. In three-dimensional cutting the relationship between the specific cutting force component in the direction of cutting speed and the specific friction force on the rake face is expressed as

E7]: Fz = M +

Ao *Seeappendix.

(1)

Ao

Dynamic Cutting Coefficients

Cutting speeds ! m/l'

Dl 'l

"I

237

Feeds ram/r, p.m.

_0,0 8

0,089 0,107 ..0~125 o,142 6 2 - 7 2 - 80 0,16 0,178 .. 0,1 9 6 0,215 93 - 1 0 6 - 1 2 3 0+25 0,285 0,316 0,36 TooL:carbide P20 ~=-6°i , ~ , = - 6 ° ; ~U=15 o

3 9 - 4 7 - 54

0,098

O

6o O

215 "10 9 F=

Ao M=

F : M+B m Ao

0

MO

1+ t g ~ ' , n

1,5 O

---~

l

M

~l~°

l° o

,

,/,o

["/.,:]

FIG. 1. Variation of the specificcomponentsof cutting force as a functionof parameter FIAo(mat. UNI C40).

where M=

M0 1 + tgy."

Now, as shown in Fig. 2(a) (X, Y) is the reference plane, normal to the cutting velocity vector V, and (AON) is the rake face of the tool, on which lies also ON, normal to the main cutting edge OA. Moreover :, is the inclination angle of the main cutting edge, 7. is the normal rake angle and ~ is the chip flow angle measured on rake face. The intersection between the 7t plane, containing both the cutting velocity vector V and the

238

V. GRASSO, S. NOTO LA DIEGA and A. PASSANNANT!

L:,', \

,%

\\

vl / I

0

,

F.x '7 /[

--.

N'

l

N \ "x

I I

\

I

\

I \I

A

Y

/ FIG. 2(a). Forces reference systems.

chip flow velocity vector Vo, and the reference plane is the ~ axis which forms the r/* angle with the X axis (feed direction). From geometric considerations the angle r/* is given by: cos(r/* - ~p) = cos ~,. cos,/

(2)

COS ~e

where the effective tool rake angle 7, in the ~ plane is determined by the relation: siny. = sin y,, cos), c o s t / + sin ~/ sin ),.

(3)

Dynamic Cutting Coefficients

~0 W

//"

239

X

/

e FIG. 2(b). Forces ref~enee systems.

The resultant cutting force R (vectorial sum of the frictional force F and of the force F'. which acts normally to the rake face of the tool) can be obtained also by summing vectorially the components R' and FN, acting respectively on the n plane and normally to it (see Fig. 2(a)). With reference to Fig. 3 we have: F = F¢ cos V=+ F= sin 7=

(4)

-F+ - .~ D + E L Ao Ao

(5)

and from equations (1) and (4)

where D=

Mo tgy,

1 + tg ~,,,,

E=

1 - Bsin~=

cos ~=

Starting from equations (1) and (5), and according to the Das and Tobias model [2], the

Po

+z

FIG. 3. Equivalent orthogonal cutting in n plane.

240

V. Gp.As$o, S. NOTO LA D m G A and A. PASSANNANTI

dynamic cutting coefficients can be derived under the following hypotheses: (I) only a straight main cutting edge is considered to be acting; (2) the dynamic component of the force R', acting on the n plane, is supposed to coincide with the dynamic component of the total force R; in other words the dynamic component of the force FN, normal to the %plane is neglected. This causes a negligible error in the range of usual values of the inclination angle 2 of the main cutting edge. DYNAMIC CUTTING

Stewart and Brown [8], using high speed photography, investigated the variation in the rake face friction and in the inclination of the shear plane in orthogonal machining under chatter conditions. They observed that normal force and friction on rake face are closely related with fluctuations of the shear plane angle during chatter and concluded that there is a direct relationship between the friction and the primary shear process in dynamic cutting. These results are in agreement with the concept that the fundamental parameter for both steady state and dynamic machining is the friction force arising on the tool-chip contact area. The ratio of this force to the uncut chip area is related in three-dimensional turning to the uncut chip thickness by an exponential law as can be seen in Fig. 4; therefore the friction force in static cutting can be written as: (6)

F/Ao = Kh o"

where K and 0~depend on the tool geometry and the cutting speed. Substituting (6) into equations (1) and (5): F, =

EMho +

(7) W

414 o

3,oo

2,00

Tool

: C<;rbide P20

C u t t i n 9 speed

1,00

0,00

~=-6

k=-6

~=t5

: 93 rn/l'

~ -1,5o

-1,oo

-0,50

Log ~toCOSq~) [mm]

FIG. 4. Variation of specific friction force against uncut thickness (mat. UNI C40).

(3,00

Dynamic Cutting Coefficients

241

But, with reference to Fig. 2(b), the uncut chip thickness on the n plane is: ho Po = cos(r/* - ¢ )

(8)

and consequently equations (7) become F= = ~

W

{Mpo cos(r/* - ¢ ) +

COS

BK[Po cos(r/*

- ~ ) l (1 -=)}

(9) F¢ = ~

W

{Dpo cos(r/* - ¢ ) + EK[p o cos(r/* - I ~ ) ] " - ' ) } .

Now, if it is assumed that the V= vector direction, the tool cutting angles and the shear plane orientation do not change during chatter, dynamic cutting components F, and F~, due to the inner modulation [(h(t) = H o sin cot)] in the direction normal to the cutting surface, can be derived. In fact, with reference to Fig. 5, the two components F 1, F 2 can be written as follows: F1[po +

p(t)] ----FI(po) +

p(t)\ dp

/o=,o

(1o)

F2[Po + p(t)] ~ F2(po) + P(t)(~-~lp=po" and the F= and F~ .components: F=(t) = F 1 - F2~

(11)

F¢(t) = F 2 + FI6 where ,=

is the angle between the instantaneous cutting direction and the cutting velocity vector. Having in mind the relations (9), the dynamic components can be written as:

F,(t) = ~

+ e(1

Aoa

~

vcos(~* -

~) (~2)

W

-,.: = cos,{[o-,-

.e°

no

t~°

"-

z

FIG. 5. Effect of the inner modulation.

V. GRASSO, S. NOTO LA DIEGA and A. PASSANNANTI

242

The dynamic components of the forces Fz and F¢ due to the outer modulation can be obtained in a similar way. With reference to Fig. 6 the segment HK indicates the instantaneous chip thickness in the plane containing chip flow velocity and cutting speed and under the same hypothesis previously proposed for the shear plane inclination, the instantaneous value of the cutting force depends on the uncut chip thickness H'K'. If the variation law of the chip thickness HK is p(t) = Ro sin cot, the thickness H'K' varies in agreement with the law p(t) = Ro sin(cot + t) where, for small amplitudes of Ro, one finds approximately

coC .~ COpoctg ~b,. e=~¢v Therefore the incremental components of the dynamic forces can be written :

Fz(t)=co-~[M+B(1-

#) ~---~-]h(t + ~ ) (13)

During chatter, when both inner and outer modulations are present with a phase angle coT = 2n(n - 0) with 0 < ,9 _< 1, the dynamic force components become:

(14)

(M+BL]ho cos

0 [L

v cos~

- ~,;

.

.

)

On the basis of the previous hypothesis (2) the dynamic component in the feed direction is

Fx(t) = F¢(t) cos ~/* and therefore: F.(t)=co~Kt[h(t)-h(t-T+

~ ) + 2tli(t)]

(15) W

z

FIG. 6. Effect of the outer modulation.

Dynamic Cutting Coefficients

243

where the stiffness and the damping due to the chatter in the directions of cutting speed and feed are expressed respectively by:

KI = M + B(1 - ~)-7-, /to

)~1 =

K1V cos(~/*

¢,) (16)

( M + B Af--~]ho cos ~/* K2=

D+E(I-0~)

cos

EXPERIMENTAL

;

)~2

KeV cos(q* - ~)

VERIFICATION

Three-dimensional dynamic cutting experiments have been carried out with two different tool geometries, employing the simplified machine tool ~ystem shown in Fig. 7. The toolholder consists of a main mass, which is linked to a ba~e by four leaf springs, and of an adjustable dynamic damper, which permits to modify the transfer function of the whole system• The direct and cross transfer function used during the tests are shown in Figs. 8 and 9. Indicating with G= = az~ + Jbx~;

G~ = a= + Jb=

respectively the direct and cross receptances, the displacement in the feed direction is given by

[io]:

X(Jm) = (a= + Jb=)F~(Jco) + (a= + Jb=)Fz(Jm )

where:

:ooL

r'no i rl

mGSS

r

FIo. 7. Tool holder [9].

V. GRASSO, S. NOTO L ^ DraG^ and A. PASSANNANT!

244

h(J03) X(J03) = cos ¢, " The Laplace transforms of the expressions (15) lead to the following equations defining the limit of stability: 1 Wt|m

= (cosfl - 1)(K2a~ + Kta,.z) +~infl(K2bxK + Ktb,~) + 03(K222b,, + Kt)~tbx=) (17)

0 = (K2ax~ + Kla=)sinfl + 03(K:22a= + K12ta,.=) + (1 - cos/~)(K2b~ + Ktb=) (18) where # = 03T- e. For given dynamic cutting coefficients Kt, K2, 2t, 22 corresponding to fixed feed and speed, the equation (18) allows to predict, for a particular frequency, the value of/~ which, after substitution in equation (17), gives a critical value of the width of cut. The corresponding values of workspeed N (rpm) are given by: N=

03

/~ + 2~m + e"

When 03is varied the unstable lobes are obtained; these lobes are close to each other and their intersections give values of W very near to the minimum value that can therefore be assumed as the W . . for the considered speed. In Table I the dynamic cutting coefficients are given as predicted by static tests for different speeds and tool geometries. Figure 10 shows the predicted stability chart for the two tool geometries employed. In this figure the experimental values of the critical width of cut are also reported.

I

~

.1o 7

2 I

-3

,

Re

I,,

-~2

-1

'0

150 Hz

130

112

-1

108 .-9

101

10E

Om3,s

85 '-3

99,s 9e,5 Hz

95

FiG. 8. Direct transf¢r function G,x.

Hz

Dynamic Cutting Coefficients

245

ITI 7 --~-" 10 -1

0

1

A

I~M~H

z i

Re I m . 107 N

0 97 95

90

-2 FIG. 9. Cross transfer function G,=.

CHATTER SENSIBILITY INDEX

The expressions (16) show that dynamic coefficients depend both on the tool geometry and on the constants M0 and B, which are characteristics of the work material. Particularly an increment of the parameter B determines small variations of dynamic stiffness in the feed direction, and more significantly increases both the damping coefficients and the stiffness in the speed direction; this effect decreases with increasing cutting speed, due to the simultaneous lowering of the friction force acting at the tool-chip interface. Therefore a material having higher value of the parameter B allows lower unconditionally stable width of cut than a material with a lower value of B. Analogously the parameter Mo, that in steady state cutting has the meaning of the minimum energy to be employed to remove the unit volume of chip, has the same influence as B on the critical width of cut. Therefore both parameters B and Mo are suitables to characterize the materials from the point of view of sensibility to chatter. It can be seen, however, that the choice of Mo has to be preferred as a chatter sensibility index because its values have a wider range of variation for different materials (from about 300 N/ram 2 for an aluminium alloy, to about 2000 N/ram 2 for a medium carbon steel); the range is certainly more limited for the parameter B (0.6 ~ 1) ['5"]. Consequently, the easily determined parameter Mo allows a materials classification both in static and dynamic cutting because it is an index of the power to be supplied for cutting and of the sensitivity to initiate self-excited vibrations. CONCLUSIONS

The comparison between the stability charts theoretically derived and those experimentally obtained confirms the validity of the dynamic model developed. In particular the obtained expressions are easy to apply and allow to predict the dynamic cutting coefficients, on the basis of the results of three-dimensional static cutting tests for different tool geometries. Moreover these expressions allow to predict the influence on the critical width of cut of the angles characterizing the tool geometry for a given work material. Among these angles the most significant are the normal rake angle and the side cutting edge whose influence can be derived from the proposed relations. In particular, by increasing the normal rake angle one obtains a higher stability, due to the decreasing of dynamic stiffness both in the cutting speed direction and in the feed direction. On the other hand, by increasing the side cutting edge angle higher values of dynamic

246

V. GRASSO,S. NOTO LA DIEGA and A. PASSANNANTI

I i i i i i i i

~

I

I

I

•

.

.

' •~

o"

~

11

o

o

o

I

o

I

A

~),,+

~L IlJ A

0 0 <

-

-

. . . . .

~ooooo~oo__

et

~

~z

~ 0 0 0

o

I

o

I

I

Dynamic Cutting Coefficients

247

E E

,),=0

f--~':5

,qo=o

/

E .o

~

0

'o//~'=-~o,;~=-6,~V=~s =

•

=_

Z

-

Q

--

o 0

0 =====~

-

0

Material

: Sl~eel. C 4 0

•

T,heoref.ical v a l u e s

TOOL

: Carbide

o

experimen~.ot vcLues

P'~O

~I = 0,15

20

40

6o

~0

100

,20

v [m/l']

FIG. 10. Critical width of cut versus cutting speed in three-dimensional cutting.

stiffness are obtained, due to the decreasing of the uncut chip thickness. The effect of the angle of inclination ), of the main cutting edge is not explicit, because it depends on the relation between this angle and the chip flow angle. Finally it is worth noticing that the same parameter which in steady state cutting classifies the work materials, can be employed in dynamic cutting as a chatter sensitivity index. REFERENCES

[U [2] [3] [4] [51 [6] [7] [8] [9] [io]

S. A. Torero, Machine Tool Vibration. Blackie, London (1968). M. K. DAs and S. A. TOBIAS,Int. d. Mach. Tool Des. Res. 7, 63 (1967). M. M. N1GM, M. M. S^DEK and S. A. TomAs,13th Int. Mach. Tool Des. Res. Conf. (1972). V. GRASSO,S. NOTO LA DmGA and M. PtAcE~rnN1, Tecnica ltaliana 2, 71 (1977). N. ALmSRTI,S. NOTO LA DmGA and A. P~SS^NN^~rrl, V Congresso Naziormle AIMETA (1980). N. ALnERTI,S. NOTO LA DmGA and A. PASSANNAm'I,Atti Accade. Sci. Lett., Palermo XXXIIi, 39 (1974). S. NOTO LA DmOA and A. PASSANNAN'rl,IV Conoresso Nazionale AIMETA (1978). V. A. STEWXWrand R. H. BROWN,Proc. 13th Mach. Tool Des. Res. Conf. (1972). V. GRASSO,S. NOTO LA DraG^ and A. PmSANNArCn,III Co~resso Nazionale AIMETA (1976). W. A. KNlorrr, Int. J. Mach. Tool Des. Res. 8, 1 (1968).

APPENDIX Starting from the results of orthogonal cutting tests with controlled contact tool [11, 12], it has been shown [6], for various speed and feed and fixed tool geometry that the chip thickness ratio depends only on the ratio

F

"rml

Ao

to

248

V. GRASSO, S. NOTO LA D~GA and A. PASSANNANTI

If the shear angle is directly related to FIA#,so is the ratio F,/Ao = (zdsin 0) (~, - cost) and also all specific cutting force components. Moreover applying the minimum work hypothesis, Merchant reasoned that the coefficient of friction on the tool rake face is independent on the shear angle; therefore he assumed a constant direction of the force between the tool and the workpiece. In paper [5] some of the authors derived the shear angle which minimizes the specific work assuming a fixed value of F/Ao on the tool's rake face and a shear stress z, directly influenced by the normal stress on the shear plane. In this way, if the specific force components F/Ao and F~/Ao are used as a coordinate system, Fig. I 1 is obtained. It is seen that the minimum work principle is satisfied for the range of cutting conditions investigated. From Fig. 11 it can be seen also that a linear relationship between F/Ao and F~/Ao holds, and consequently it can be written: Fz

F

Ao

Ao

~=M+B~.

(1)

The parameter M is the minimum energy that must be employed to remove the unit volume of chip and consequently it must be a characteristic parameter of work material that will vary only with tool geometry. Particularly if F/Ao ffi 0 one must have ~ = 45 ° + ~, because the normal to the tool rake face is a direction of a principal stress; in this case, as can be seen from Fig. 12 one obtains easily (2)

(FdAo)mi, = M = 1 + tg:~ ~

.

3.1o 9

2

3

I

i

0

t

I

,

i

2

l

l

Straiqht Line regression

i

t

3'109

~

A.

FIc. 11. Relationship between FdAo and F/Ao (mat. SAE 4135).

FIG. 12. Specific forces system in orthogonal cutting (F/Ao = 0).

-~.

Dynamic Cutting Coefficients

249

Q

In practice, for a wide range of cutting conditions, the shear stress on the shear plane remains in a very narrow range and therefore, when F/A o = 0, z, can be assumed to be constant, indepeodent on tool geometry. In this hypothesis the relationship (I) can be written F,

M o

F

Ao = I + tg~, + B ~ .

(3)

The validity of (3) has b ~ n tested in three-dimensional cutting [7], varying the tool geometry and using Fisher's test. It has bc~n found that this relationship holds if the rake angle is defined in a section normal to the cutting edge. REFERENCES Ill] [12]

H. TAKEYAMAand E. UsuI, Trans. ASME 1089-1096 (1958). B.T. CHAO and K. J. TRIC,OER, Trans. ASME. Set. B. J. Eng. Ind. 139-151 (1959).