Closed form formulation of cutting forces for ball and flat end mills

Pergamon Int. J. Maeh. Tools Manuf~t. Vol. 37, No. 1, pp. 17-27, 1997

Copyright © 1996 El~vier Science Ltd Printed in Great Britain. All right.~ reserved

0890.-6955197517.00 + .00

PII: $0890-6955(96)00018-1

C L O S E D F O R M F O R M U L A T I O N OF C U T H N G F O R C E S F O R

B A L L A N D F L A T E N D MILLS

F. ABRARI* and M. A. ELBESTAWI*t

(Original received 22 July 1995; in final form 16 January 1996)

Abstract In this paper, a set of basis functions for the calculation of cutting forces in milling are introduced. It is shown that the cutting force at any tool position can be determined by the linear combination of the force basis functions. The method is based on the projection of the chip load area onto the reference coordinate planes. Due to the analytical integration of the cutting forces along the cutter edge, the developed closed form equations provide a fast means of calculating the cutting forces. The validity of the method is experimentally verified for both fiat and ball end mills. Copyright © 1996 Elsevier Science Ltd

NOMENCLATURE

Fx, Fr, Fz Ax, At, Az x , Y , Z R f K i Otn

0

4>

cutting force components in X, Y, Z directions projections of the chip load area onto YZ, XZ and XY planes reference system of coordinates radius of ball or flat end mill feed per tooth (mm/tooth) specific cutting force matrix (N/mm 2) inclination angle of ball end mill flute normal rake angle of ball end mill sin (ctn) angle of cutting edge of a ball end mill in spherical coordinate helix angle of flat end mill rotational position of the tool rotational position of ball end mill edge in spherical coordinates

1. INTRODUCTION

In the first rigorous geometrical analysis of the milling process, published by Martelloti in the early 1940s [1, 2], it was shown that the average power consumed in milling is primarily a function of the average undeformed chip thickness. Later, Koenigsberger and Sabberwal [3, 4] also observed that there is a strong relation between the instantaneous chip thickness and the tangential cutting force (also called the power component). The following simple equation has been widely used by various researchers:

F, = K, bt = K,A (1)

where K, is the specific cutting pressure, t is the undeformed chip thickness and b is the width of the cut. A=bt is the chip load area. To compute the undeformed chip thickness t at any instant, one needs to establish a relationship for the distance between the tool paths of current and previous teeth.

In all of the cutting force models developed for milling (see for example [5-9]), two fundamental assumptions are made:

(1) The kinematics of the milling operation can be modeled by decoupling the motions of the spindle (tool) and table (workpiece). The immediate conclusion from this assumption is that the cutter path seen by an observer located on the workpiece is circular. Consequently, the undeformed chip thickness can be approximated by [3, 4]:

*Mechanical Engineering Department, McMaster University, Hamilton, Ontario, Canada L8S 4L7 tTo whom correspondence should be addressed.

17

18 F. Abrari and M. A. Elbestawi

t = fsin(~b) (2)

wherefis the table's advancement per spindle rotation and th is the spindle's rotational position measured from such an axis that satisfies the boundary conditions of t at the exit and entrance positions.

(2) The mechanics of machining of any complex process can be modeled by an aggre- gation of oblique cuts, if and only if the cutting edge is divided into infinitesimal elements [10]. The immediate application of this assumption is to discretize the tool into thin slices, as shown in Fig. 1, calculate the cutting force applied to each thin slice of the discretized tool and then sum the differential cutting forces up for all the slices and teeth engaged along each coordinate direction. Software using such a numerical integration would be inherently slow and practically inadequate for real time calculations of cutting forces.

To gain higher speed in the calculation of cutting forces, some researchers have explicitly integrated the expressions of cutting forces for fiat end mills. Although the resulting closed form formulation of cutting forces introduced here for fiat end mills is essentially identical to those of direct integration derived in [11, 12], the physical interpret- ation of our approach is totally different.

2. GENERAL CONCEPT OF CLOSED FORM SOLUTION

The purpose of this study is to provide a method for fast calculation of cutting forces in real time. The goal is achieved by eliminating the numerical integration of the cutting forces through introducing a set of force basis functions. To develop the required force basis functions, the chip load area in any tool position is projected onto the reference coordinate planes. Accordingly, instead of having the chip load area cutting the workpiece one may think that three mutually projected areas are doing the cutting operation. Thus, as the projected areas navigate through the workpiece, a pair of normal and frictional forces will be applied to each projected area, in such a way that the components of the cutting force in any coordinate direction, e.g. X, will be associated with:

(1) A normal force on Ax. (2) Two frictional forces along the X direction on the other two projected areas, i.e. Av

and Az [13]. The same procedure is used for the Y and Z coordinate directions.

Hence, the components of the cutting force in any tool position can be written by linear combination of the projected areas in that tool position. Mathematically:

{F} = [K]{A} (3)

where {F} is the total cutting force vector at any tool position, {A} is the chip load vector

/ /

Cutting edge

Discretizing planes

Fig. 1. Discretized cutting edge.

Closed Form Formulation of Cutting Forces for Ball and Flat End Mills 19

at that tool position and [K] is the matrix of specific pressures. It is clear that through this approach the numerical integration of cutting forces, which is computationally expens- ive, is replaced by analytical integration. In the next two sections the method is applied to flat and ball end mills.

3. CLOSED FORM FORMULATION OF THE CUTTING FORCE EQUATIONS FOR FLAT END MILLS

Figure 2 shows the peripheral milling operation of a conventional fiat end mill with radius R and helix angle/3. Figure 3 shows an infinitesimal chip load area and its projections on the reference coordinate planes associated with the process shown in Fig. 2. In Fig. 3, f is feed per tooth and ~b is tool rotational angle. For a fiat end mill with helix angle /3 one has: R dq~/dy = tan/3 or dy = R d~b/tan/3. From Fig. 3, it is clear that the projection of chip load area on the X coordinate plane is:

1 1 1 dA~ = ~f[sin(~b)cos(~b) + sin(d~ + d~b)cos(th + dth)].dy = ~ f [ } sin(2th)

1 + ~ sin(2~b) + dq~cos(2$)].R dth/tan(/3)

(4)

By neglecting higher order differentials:

1 d + dA~=~fR[sin(2~)] tan/3" (5)

Similarly it can be shown that:

Z

Fig. 2. Peripheral operation of a conventional flat end mill.

i

Fig. 3. Chip load projections of peripheral milling.

x


dAr = Rfsin(~b) de#

1 dth dAz = -~ Rf(1 -cos(2~b)). tan-~ " (6)

As discussed earlier, the cutting force components in each direction could be predicted by a linear combination of the projected areas, mathematically:

F Kxz]f Axl dFzJ Kzr K~zJtdA~J

(7)

The cutting force components generated in peripheral milling, of a cutter with helix angle/3 are shown to be [12]:

gf dFx - 4tanfl

rbkr (1-cos2~b)] d~b - - - [2K, sin24~ + 2

dFy = K, Rfsin~b d~ (8)

Rf dFz - 4tan/3

wk~ - - - [2K,(1-cos2~b) + 2 ~ sin2~b] d~b.

Comparing Equations (7) and (8) results in the following:

1 ~r ~ - ~ K , 0 ~ K , -

K = R f OK, O

fir 1 2s~n~ Kr 0 - ~ Kt

(9)

where K, is used to compute the tangential and 7/r and Kr are used in the calculation of the radial component of the cutting forces including the ploughing force. The above K matrix could be used as the specific force matrix.

Integrating both sides of equation (7) leads to the simple equation {F} = [K] {A}, where:

Ax 4tan/3 i__~'l (cos2~b,.,-cos2tk~xt)j

N Ar = Rf ~ (COS~ent--COS~ext)j

j=l (lO)

Az - 4tan/3 j= i [2~b, xt + sin2~b~,-(2~b~n, + sin2tk¢x,)]j

where ~be,, and ~bex, are the entrance and exit angles along the cutting edge and N is the number of teeth engaged.


4. CLOSED FORM FORMULATION OF THE FORCE EQUATIONS FOR BALL END MILLING

Figure 4 shows a perspective from the chip load area of a typical ball end mill. Like any other complex tool, the chip load area of a ball end mill is a twisted strip in 3D space. The edge coordinates of a typical ball end mill are [7]:

X = Rcosa.sinO

Y = Rcosa.sina.(1-cos0) (11)

Z = -R(cos2ancosO + sin2an).

As the spindle rotates, the following equation can be used to map the coordinates of the reference edge position on the new ones. ~ is the rotational position of the spindle.

= - s i n ~ cos4~ 0 / Y (12)

0 0 1 J Z

The rotational position of any point on the rotated cutting edge can be defined by: = tan-l(y/x) and determined by the expansion of the above equation:

0 tan(O + ~b) = 6 tan ~ (13)

where 6 = sin(a,). Having obtained the tool rotational position ~b, and the cutting edge rotational angle ~ , one can compute the projection of the chip load area on the coordinate planes. For example, dAx can be determined as follows:

dAx = fsin2q~lz = fsinh0[R(1 - 62)sin(0)d0]. (14)

Computing sin2(qJ) using equation (13) and substituting in equation (14) will result in the following:

O_.q tan

dA~ = 2Rf( 1 - 8 2) dO (15)

~ R e Y ference

Fig. 4. Perspective of the chip load area for a typical ball end mill.


where 77 = tan(t0). Integrating equation (15) along the cutting edge yields:

6 Ax = 4 R f ( 1 - ~ ) ~ I~

i= l

where

1 11- 2( 1 - 82)

(32cos2to-sin2to).cos2(O/2)

3 12 - 4(1-81) sin(2to).(O +sinO)

13 = - ~ cos(2to)-log(cos(0/2))

3 I4 - 2(1-32) 2 sin(2to).0

1( )2 15 = - ~ ~ cos(2to).log(1 + 32tan(O/2))

3 2 16 = ( 1 ~ - ~ ) s i n ( 2 t o ) ' t a n - 1 (Ttan(0/2)) .

Similarly [13]:

dAv = 2Rf( 1 - 8 2)

0 0 0 - ~ t a n 2 + 3(1-r/2)tan 2 ~ + 32~tan 3

d0

where again "0 = tan(to). Integrating the above leads to:

6 Ar = 2Rf(1 - 82) ~ li

i=1

where

1 +32 Ii - 4(1 - 3 z) sin(2to)'c°s2(O/2)

3 12 - 4(1-32) cos(2to).(0 + sin0)

3 2 ,3_-

3 14 - 2 ( 1 - 3 2) cos(2to).0

(16a)

(16b)

(17)

(18a)


1(,)2 I~ = - ~ ~ sin(2~b).log(1 + ~2tan2(0/2))

2

,6- (18b)

Similarly [13]:

1 2 2 tan(~b + ck)sin(~b) d$ + ~ f sin ($)d~b. dAz = 2R3'3 82 + tan2(~ b + ~b) (19)

Integrating the above gives:

1 2 Az = 2Rf~[llcostk + 12sintk] + ~ f .13 (20a)

where

1 sin(0+ qb)- - - . tan -~ I , - 1_62 1_82 8~/-~_8z ) ~ / ~ - s i n ( q t + 4))

1 1 ( lx/~--~cos(d/+ tk)) (20b) 1 cos(~+~b)+ _~52 - - tanh -t t2 - 1 - 8 2 i - - - - "

1 1 13 = } qJ- ~ sin(2qJ).

Here again, for a given tool/workpiece and cutting conditions there exists a matrix of specific pressures [K] such that {F} = [K]{A} best fits into experimental data. Vector {A} is given by Equations (16), (18a) and (20a).

5. EXPERIMENTAL SETUP AND VERIFICATION

Two separate programs were developed to simulate the milling operations of flat and ball end mills using the developed equations. To verify the validity of the method and existence of a specific pressure matrix as defined in Equation (3), the experimental data were compared with the simulation results. The experiments were conducted on a YAM NC 3-axis milling machine center with 7.5 kW spindle power. A table dynamometer was used to measure the cutting forces in the X and Y directions. Four piezoelectric force transducers, two in the X and two in the Y direction were mounted on the dynamometer. A microcomputer pc AT (Intel 386) and a 12 bit, 16 channel A/D board (DT2821) were used in the data acquisition. To ensure an equal number of acquired data points per rotation of the tool, an asynchronized external clock of 1024 pulses per rotation was used. The workpiece material was aluminum 7075-T6.

Figure 5 compares the experimental and simulated cutting forces of peripheral up milling of a 4 flute HSS flat end mill. The tool diameter was 25.4 mm and its helix angle was 30 °. The depth of cut and the immersion ratio were 5.2 and 0.13 mm respectively. Figure 6 was obtained using a 2 flute HSS ball end mill in up milling operation. The immersion ratio in this case was 0.5 and the constant depth of cut of 5.2 mm was positioned relative to the ball end mill such that the workpiece's bottom was 5.5 mm above the bottom of the tool. Both tests were done without using coolant. Comparison of the experimental data MII~ 37-1-~


-100

, - , -200

Z V

X -.300 It,

-400

-500

/

|

~, experlra#ntal

i i i i

simulation

Down Milling, Spindle speed : 600 rpm, 4 flute flat end mill

I I ,, I I I I I I I

0.02 0.04 o.oe 0.08 0,1 0.12 0,14 0,16 0.18 Time ( sec )

0 0.2

450 . . . . . . . . . 1 Down Mil l ing, Spindle speed = 600 rpm, 4 flute flat end mil l 4oo 35C

Gn l I

300 simulation

Z 250

L~ 200

150

100

5O

o "J

I | I I

-5°0 0.02 0.04 0.06 0.08 01.1 0.112 0.14 0.16 0.118 • 0.2 Time ( sec )

Fig. 5. Comparison of simulation results with experimental data--fiat end mill.

with simulation results reveals that the developed method is capable of generating cutting force patterns quite similar to experimental data.

6. DISCUSSION

In the modeling approach described in this paper, the machining process is divided into two main aspects, geometrical and mechanical. The geometrical aspects of machining are grasped by the basis functions in vector {A } and all other complicated mechanical aspects of machining such as the effect of the cutting temperatures, cutting speeds, rake angles, coolant and so on are combined in the specific pressure matrix [K].


ZL, oo v

60O ! i |

5O(

40(

30C

IO0

0

-100 ~ g/as=/ag/oal

I I I I

-2000 0.02 0.04 0.06 O.J

exiper/mermd /

i !

Up Milling.Spindlespeed=6OOrpm.2flutcbidl endmill I - • I I I I

0,06 0.1 0.12 T ' ~ e ( scc )

0.14 0.16 I

0.18 0.2

1

I I

0 0.02 0 .04 0.06

experimental

I I I I l ,

o.o8 0.~ 0 J 2 o r 4 0.t6

Time ( sex: )

V ~

I

0.18 0.2

Fig. 6. Comparison of simulation results with experimental data-ball end mill.

It is important to note that the components of the K matrix in the closed form solution are average values as opposed to spatial specific cutting pressure values. In the latter case each differential element on the cutting edge has its own specific pressure value that must be used in the calculation of the cutting forces associated with that particular element. However, in the closed form solution a single K, value is assigned for the entire engaged cutting edge.

In the Rake Face Force Model developed by Endms et al. [14], the shearing forces are modeled in the rake face using a normal rake face force coefficient K~r and a rake face coefficient of friction /.ta instead of the specific cutting and thrust energies. A similar


formulation will be used here in the formulation of cutting forces based on the calculated projected areas.

Figure 7 shows the chip load projections as well as their direction of motion during the up milling process of a ball end mill. Due to the motion of At and Az in the X direction there will be a frictional force in the X direction applied to them. The magnitude of these frictional forces depends on the friction coefficient as well as the normal forces of Fr and Fz. Suppose the coefficients to be/xy and/Xz, then the friction forces on Ar and Az in the X direction will be:

Ixr'Fr = Ixr'KrrAr = Kxr'Ar

/xz.Fzcos(~b) = tzz'KzzAzcos( ~b) = Kxz'Azcos( tp). (21)

These two frictional components have to be summed up with the normal force Fx = KxxAx to give the total force applied to the tool in the X direction. The above con- dition is true for the total force in the Y direction as well. However, for the total cutting force in the Z direction and since none of the Ax and Ars have motion in the Z direction, there will be no frictional force associated with them in the Z direction (as shown in Fig. 8). This causes Kzx and Kzr to be zero in the specific cutting pressure matrix [K] of an operation in which there is no feed in the axial direction.

7. CONCLUSION

The developed closed form basis functions for the cutting forces in the milling process show excellent capability of matching the experimentally observed cutting force pattern. The method provides a new insight to the nature of the end milling process by replacing the chip load area with three mutually perpendicular planar areas. The size of each of the areas is determined by projecting the chip load area onto the corresponding coordinate plane. As the three mutually perpendicular areas advance through the material, the components of the developed cutting force are computed by adding the normal and frictional forces exerted on the projected areas. Due to analytical integration of cutting forces along the cutting edge, this method is much faster than those using numerical integration tech- niques. The method is also potentially applicable to other types of cutting processes where some geometrically definable chip load area is involved, such as drilling and face milling.

Z

x

Fig. 7. Chip load projections in ball end milling (up milling).

Closed Form Formulation of CuRing Forces for Ball and Flat End Mills 27

Z

~ Y

A Y

Fig. 8. Chip load projections in Z-direction.

REFERENCES

[1] M.E. Martelloti, An analysis of the milling process, Trans. ASME 63, 677-700 (1941). [2] M.E. Martelloti, An analysis of the milling process, part II, down milling, Trans. ASME 67, 233-251 (1945). [3] F. Koenigsberger and A.J.P. Sabberwai, Chip section and curing force during the milling operation, Ann.

CIRP X (1960). [4] F. Koenigsberger and A.J.P. Sabberwal, An investigation into the curing force pulsation during milling

operations, Int. J. Machine Tool Des. Res. X (1961). [5] C. Sim and M. Yang, The prediction of the curing force in bali end milling with a flexible cutter, Int. J.

Machine Tools Manufact. 33, 267-284 (1993). [6] G. Yucesan and Y. Aitintas, Mechanics of ball end milling process. PED Voi. 64, Manufacturing Science

and Engineering, pp. 543-551, ASME (1993) [7] M. Yang and H. Park, The prediction of cutting force in ball end milling, Int. J. Machine Tools Manufact.

31, 45-54 (1991). [8] M.A. Elbestawi and R. Sagharian, Dynamic modelling for prediction of surface errors in the milling of

thin walled sections, J. Materials Processing Technol. 25, 215-228 (1991). [9] W.A. Kline, R.E. DeVor and J.R. Lindberg, The prediction of cutting forces in end milling with application

to cornering cuts, Int. J. Machine Tool Des. Res. 22, 7-22 (1982). [10] M.C. Shaw, N.H. Hook and P.A. Smith, The mechanics of three dimensional curing operations, Trans.

Am. Soc. Mech. Engrs 100, 222 (1979). [11] Y. Altintas and A. Spence, End milling algorithms for CAD systems, Ann. CIRP 40, (1991). [12] S. Jain and D. C. H. Yang, A systematic force analysis of the milling operation, Proc. ASME, Winter

Annual Meeting, San Francisco, pp. 55-63 (1989). [13] F. Abrari, A regenerative dynamic force model for ball end milling, Master's thesis, McMaster University,

Hamilton, Canada (1994). [14] W. J. Endres, R. E. DeVor and S. G. Kapoor, A dual-mechanism approach to the prediction of machining

forces; part l, model development and calibration, PED Vol. 64, Manufacturing Science and Engineering, pp. 563-576, ASME (1993).

Documents

Closed form formulation of cutting forces for ball and flat end mills