modeling of cutting forces in end milling operations

8/6/2019 Modeling of Cutting Forces in End Milling Operations

http://slidepdf.com/reader/full/modeling-of-cutting-forces-in-end-milling-operations 1/8

Tamkang Journal of Science and Engineering, Vol. 3, No. 1, pp. 15-22 (2000) 15

Modeling of Cutting Forces in End Milling Operations

Wen-Hsiang Lai

Department of Mechanical Engineering

University of Kansas

Lawrence, KS 66045

E-mail: [email protected]

Abstract

According to previous research of dynamic end milling models,

the instantaneous dynamic radii on every cutting position affects the

cutting forces directly since the simulated forces are proportional to

the chip thickness, and the chip thickness is a function of dynamicradii and feedrate. With the concept of flute engagement introduced,

it is important to discuss it with respect to radial and axial depths of

cut because the length of the engaged flutes is affected by factors in

the axial feed and rotational directions. Radial and axial depths of cut

affect the “contact area”, which is the area that a cutter contacts with

the workpiece. When radial and axial depths of cut increase, the

cutting forces also increase since the engaged flute lengths are

increased. Therefore, in order to have a clearer idea of the milling

forces, the influences of dynamic radii, cutting feedrate, and radial and

axial depths of cut are discussed in this paper.

Key Words: simulated forces, milling forces, dynamic radii, cuttingfeedrate, flute engagement, radial and axial depths of

cut, rake angle

1. Introduction

Milling operations are one of the most commonmachining operations in industry. It can be used for

face finishing, edge finishing, material removal, etc.

There are several parameters that influence the

forces acting on the cutter. Because of these

parameters, the forces may become unpredictable

and result in larger dimensional variations when

products are produced.

In order to discuss these dimensional variations,

several flexible simulation models have been

introduced [1, 7, 10, 11, 14, 16, 17]. In these

flexible models, simulated forces are proportional to

the “chip thickness”— Tc. The chip thickness refers

to the thickness of material that each flute on a cutter

removes at a certain position. The chip thickness is

expressed as a function of dynamic radii and feedrate.

Therefore, the dynamic radii and feedrate are key

factors influencing simulated forces. With the

concept of flute engagement introduced, it is

important to discuss it with respect to radial and

axial depths of cut since the flute engagement is

affected by factors in the axial feed and rotationaldirections.

“Flute engagement” refers to the flutes on a

cutter that are engaged at any instant in time as a

milling operation is performed. The role of flute

engagement in milling operations is important

because it affects not only the cutting forces, but also

the cutting surface. However, flute engagement is

influenced by the radial and axial depths of cut

because radial and axial depths of cut affect the

width and length of the “contact area” in the axial

feed and rotational directions, respectively. That is,

the deeper the radial or axial depths of cut, the more

flutes will be engaged, and thus increase the length

of the engaged flutes. Figure 1 shows the geometry

of the “contact area”.

Figure 2 shows the “flattened” representation of

a cutting tool. On the diagram, a horizontal and a

vertical line are drawn from the lower left corner.



Tamkang Journal of Science and Engineering, Vol. 3, No. 1 (2000) 16

The length of the horizontal line is CL, the circular

length of the contact surface between the cutter and

the workpiece, and AL is the distance between eachflute in the axial direction. The length of the vertical

line is equivalent to the axial depth of cut, a. This

diagramming technique will be used later in the

paper to describe the effects of radial and axial

depths of cut. Another important influence on

milling operations is the influence of rake angle.The concept of rake angle will be also discussed in

this paper.

Contact area

Workpiece

Cutting tool

Figure 1. The geometry of the contact area

AL

a

CL

Figure 2. “Flattened” representation of cutting tool

2. Influences of Dynamic Radii

In a simulation model, the most significantinfluence on peripheral milling forces is the chip

thickness. However, the chip thickness is mainly

affected by dynamic radii, which are caused by

cutter runout and tilt. In [17], Sutherland and DeVor

indicate that the dynamic radii caused by cutter

runout and tilt are:

RAD(i,k)=[RAD²+ρ²+2RADρcos(λ-δλ-2π(k-1)/Nf)+

(PL-Z[i])²sin²(τ)+2(PL-Z[i])sin(τ)(ρcos(φ)

+RADcos(λ-δλ-2π(k-1)/Nf-φ))]1 2 /

(1)

where

ρ: cutter offsetφ: locating angle for the end mill tilt

λ: locating angle for cutter offset

δλ: measured angle

τ: end mill tilt angle

Z[i]: height above the free end of the cutter

Nf : number of flutes

PL : effective length of cut

β(i,j,k): flute engagement angle on the ith disk,

jth angular position and kth flute

Therefore, when φ ≈ 0, and cos²(λ-δλ-2π(k-1)/Nf )≈1

(or λ-δλ-2π(k-1)/Nf ≈ 0 or π), equation (1) can be

rewritten as:

RAD(i,k)=RAD+[ρ+(PL-Z[i])sin(τ)]cos(λ-δλ-2π(k-1)/ Nf)

=> RAD(i,k) =RAD+ξ cos (λ -δλ-2π(k-1)/Nf) (2)

where ξ = ρ + (PL - Z[i]) sin (τ) (3)

According to equation (2), the dynamic radii for

each flute are shown in Figure 3 within one

revolution of the tool.

0.3735

0.374

0.3745

0.375

0.3755

0.376

0.3765

0 1 0

2 5

3 5

5 0

6 0

7 5

8 5

1 0 0

1 1 0

1 2 5

1 3 5

1 5 0

1 6 0

1 7 5

1 8 5

2 0 0

2 1 0

2 2 5

2 3 5

2 5 0

2 6 5

2 7 5

2 9 0

3 0 0

3 1 5

3 2 5

3 4 0

3 5 0

Degrees

D y n a m i c r a d i u s ( i n

c h e s )

(a) Tooth 1



Wen-Hsiang Lai: Modeling of Cutting Forces in End Milling Operations 17

0.3735

0.374

0.3745

0.375

0.3755

0.376

0.3765

0 1 0

2 5

3 5

5 0

6 0

7 5

8 5

1 0 0

1 1 0

1 2 5

1 4 0

1 5 0

1 6 5

1 7 5

1 9 0

2 0 0

2 1 5

2 2 5

2 4 0

2 5 5

2 6 5

2 8 0

2 9 0

3 0 5

3 1 5

3 3 0

3 4 0

3 5 5

Degrees

D y n a m i c r a d

i u s

( i n c h e s )

(b)Tooth 2

0.3735

0.374

0.3745

0.375

0.3755

0.376

0.3765

0 1 0

2 5

3 5

5 0

6 0

7 5

8 5

1 0 0

1 1 0

1 2 5

1 4 0

1 5 0

1 6 5

1 7 5

1 9 0

2 0 0

2 1 5

2 2 5

2 4 0

2 5 5

2 6 5

2 8 0

2 9 0

3 0 5

3 1 5

3 3 0

3 4 0

3 5 5

Degrees

D y n a m

i c r a d i u s

( i n

c h e s )

(c) Tooth 3

0.3735

0.374

0.3745

0.375

0.3755

0.376

0.3765

0 1 0

2 5

3 5

5 0

6 0

7 5

8 5

1 0 0

1 1 0

1 2 5

1 4 0

1 5 0

1 6 5

1 7 5

1 9 0

2 0 0

2 1 5

2 2 5

2 4 0

2 5 5

2 6 5

2 8 0

2 9 0

3 0 5

3 1 5

3 3 0

3 4 0

3 5 5

Degrees

D y n a m i c r a d i u s

( i n c h e s )

(d) Tooth 4

0.374

0.37425

0.3745

0.37475

0.375

0.37525

0.3755

0.37575

0.376

0 1 0

2 0

3 5

4 5

6 0

7 0

8 5

9 5

1 1 0

1 2 0

1 3 5

1 4 5

1 6 0

1 7 0

1 8 5

1 9 5

2 0 5

2 2 0

2 3 0

2 4 5

2 5 5

2 7 0

2 8 0

2 9 5

3 0 5

3 2 0

3 3 0

3 4 5

3 5 5

Degrees

D y n a m i c r a d i u s

( i n c h e s )

(e) 4 flutes

Figure 3. The dynamic flute radii within one revolution for a 4-flute, 3/8 inch diameter tool, i.e. Nf =4, RAD = 0.375 inch.

“One revolution” refers to the angle (δλ) measured back as the flute engagement wraps up the helix angle.

From equation (2), it is obvious that dynamic

radii are expressed as a function of the axial cutting

length (Z[i]), instead of a function of the rotational

angle. From another point of view, by considering

the effect of cutter offset (ρ) and the measured

angle (δλ), the cutter radius is modified with

respect to the position of engaged flutes and cutter

offset; that is, the cutter radius varies at any

moment not only because of the flute position

(caused by δλ), but also because of the cutter offset

(caused by ρ). Figure 3 (e) shows the dynamic

radii varying between RAD+ξ and RAD-ξ. Inaddition, Figure 3 also shows that when flute 1 and

flute 3 reach the maximum radii, flute 2 and flute 4

become the minimum radii respectively because the

spacing angles between them equal 180°.

ξ

ξ




3. Influences of Feedrate

In order to assume the path that every flute cuts isa circle, it is necessary to assume the feedrate (f) ismuch smaller than the radius. However, if the

cutting speed in the x-direction is low and the

spindle speed (N) is high, f t (feed per flute) will

become very small and fit the above assumption.

In [8, 17], tangential forces (DFTAN), radial

forces (DFRAN) and chip thickness are expressed as:

DFTAN = Kt × Dz × Tc (4)

DFRAN = Kr × DFTAN (5)

Tc(i,j,k)=RAD(i,k)-RAD(i,k-m)+m×ft×sinβ(i,j,k) (6)

where m represents flute k is removing the materialleft by flute k-m (explained in [8]), Kt and Kr are the

coefficients of tangential and radial force equations,

and Dz is the chip thickness in axial direction.

Therefore, it is obvious that Tc on the ith disk, jth

angular position and kth flute is a function of dynamic

radius − RAD(i,k) and feedrate.

When the feedrate is increased, the chip thickness

is increased instantaneously as expressed by equation

(7), and the tangential forces increase because the

forces are proportional to the chip area (Dz × Tc).ft = f / (N × Nf) (7)

Figure 4 shows four different measured forces with

respect to four different increasing feedrate. Each

cycle on the graphs correspond to a single revolution

of the tool. As the feedrate increases, the magnitude

of the forces increase correspondingly. Also, theeffect of dynamic radii increase with the feedrate as

exhibited by higher force variations within a cycle.

0

10

20

30

40

50

60

1

2 6 9

5 3 7

8 0 5

1 0 7 3

1 3 4 1

1 6 0 9

1 8 7 7

2 1 4 5

2 4 1 3

2 6 8 1

2 9 4 9

Samples

Y F o r c e s ( l b )

(a) f = 1 (in/min.)

0

10

20

30

40

50

60

1 324 647 970 1293 1 616 1 939 2262 2 585 2 908

Samples


(b) f = 2 (in/min.)

0

10

20

30

40

50

60

1 324 647 970 1293 1 616 1 939 2262 2585 2 908

Samples


(c) f = 3 (in/min.)

0

10

20

30

40

50

60

1 329 657 985 1313 1641 1 969 2297 2 625 2 953

Samples


(d) f = 4 (in/min.)

Figure 4. Simulated forces with respect to four

different feedrate (Nf = 4, RAD = 0.375)

4. Influences of Radial and

Axial Depths of Cut

Flute engagement in the milling forces isimportant because it influences the forces directly.

However, it is also influenced by the radial and axialdepths of cut because radial and axial depths of cut

affect the width and length of the “contact area” in

the feed and rotational directions, respectively. That

is, the deeper the radial or axial depths of cut, the

more flutes will be engaged, and thus the lengths of

the engaged flutes.

4.1 Radial depth of cut

The radial depth of cut plays an important role in

milling forces because as the radial depth of cut is

increased, the “contact area” increases in therotational direction, and the forces becomes larger.

Figure 5 shows the effect of different radial depths of

cut on the width of the contact area. When the radial

depth of cut increases, the width of the “contact

area” is increased, and thus increases the forces.




CLcutter circular length

PL

FSL

flute 1flute 2flute 3flute 4

a

(a) Radial D.O.C. = 33%

cutter circular length

PL

CL FSL

a

(b) Radial D.O.C. = 50%

cutter circular length

PL

FSL

a

CL

(c) Radial D.O.C. = 66%

Figure 5. Representation of a cutting tool

with respect to different radial D.O.C.

0

10

20

30

40

50

1 314 627 940 1253 1 566 1 879 2 192 2505 2 818

Samples


(a) Radial D.O.C. = 33%

0

5

10

1520

25

30

35

40

45

50

1

2 1 0

4 1 9

6 2 8

8 3 7

1 0 4 6

1 2 5 5

1 4 6 4

1 6 7 3

1 8 8 2

2 0 9 1

2 3 0 0

2 5 0 9

2 7 1 8

2 9 2 7

Samples

Y f o r c e s ( l b )

(b) Radial D.O.C. = 50%

0

5

10

15

20

25

30

35

40

45

50

1

2 2 3

4 4 5

6 6 7

8 8 9

1 1 1 1

1 3 3 3

1 5 5 5

1 7 7 7

1 9 9 9

2 2 2 1

2 4 4 3

2 6 6 5

2 8 8 7

Samples

Y f o r c e s ( l b

)

(c) Radial D.O.C. = 66%

Figure 6. Simulated forces with respect to different

radial depths of cut (1 sample ≈ 0.2 degree)

From Figure 6, it demonstrates that when the

radial depth of cut increases, milling forces increase

since the “contact area” is increased. In addition, it

also shows that the shape of the measured force

charts becomes smoother when radial D.O.C.

increases. This is due to the flute engagement

because when the radial D.O.C. is 50% or 100%, the

engaged length is constant. This is because when

one flute enters the workpiece, the preceding fluteexits the workpiece simultaneously.

4.2 Axial depth of cut

Axial depth of cut is another factor influencingthe “contact area” since it affects the axial cutting

length of the area. That is, when the axial depth of

cut is increased, the length of engaged flutes

increases, and the milling forces also increase.

Figure 7 and 8 show that when the radial depth of cut

is fixed (<50%), the effect of different axial depths

of cut on the length of the “contact area” increases.

When the axial depth of cut increases, the length of

the contact area is increased, and thus the forces.

AL


PL

FSL

flute 1 flute 2 flute 3

flute 4

a

C

A

(a) a < C




AL


PL

FSL

a

A

C

(b) a ≥ C

Figure 7. Representation of a cutting tool

with respect to different axial D.O.C.

0

5

10

15

20

25

30

1 299 597 895 1193 1491 17892087 2385 2683 2981

Samples

Y F

o r c e s ( l b )

(a) a = 0.25 in.

0

5

10

15

20

25

30

1 296 591 886 1181 1476 1771 2066 2361 2656 2951

Samples


(b) a = 0.5 in.

Figure 8. Simulated forces with respect

to different axial depths of cut

From Figure 7, it should be noted that if the axialdepth of cut is less than AL (a < C) and the radial

depth of cut is fixed, when the axial depth of cut is

deeper, the forces will be larger. On the other hand,

if the axial depth of cut is larger than AL (a ≥ C),

more flutes will be engaged and thus resulting inincreased forces. FSL, CL, and AL can be expressed

as:FSL = 2π × RAD / Nf (8)

r/RAD)(1cosRAD=CL 1 −× − (9)

AL = FSL / tan(αhx) (10)

where αhx is the helix angle.

When the values of cutting forces are measured

(with positive x in direction of feed, and positive y

into the material), x-forces are changed from

negative values to positive values when the radial

depth of cut is increased, while y-forces are always

positive. Figure 9 (a) shows the force analysis in x

and y directions with the radial depth of cut much

less than the radius (<< 50%). In this case, only oneflute is engaged in the cut, and the y-forces are

always in the positive y-direction and x-forces are in

the negative x-direction. Figure 9 (b) shows the

radial depth of cut larger than the radius (>50%). In

this case, two flutes (more than one flute) are

engaged in the cut, and the y-forces are also in thedirection of positive y-direction, and x-forces are

always kept in positive x-direction.

radial forcetangential force

Fr (x)Ft (x)

Fr (y) + Ft (y)

x (positive)

y (positive)

(a) Radial depth of cut is less than 50%

radial force

tangential force

Fr (y)

Ft (y)

Fr (x) + Ft (x)flute 1

flute 2

radial forcetangential force

(b) Radial depth of cut is more than 50%

Figure 9. The geometry of total forces in X and Y

directions

From another point of view, when the radial

depth of cut is less than 50%, the total forces acting

on the engaged tooth load the workpiece in the

negative x-direction and positive y-direction. Whenthe radial depth of cut is greater than 50%, the total

forces load the workpiece in the positive x-direction

and positive y-direction.

5. Rake angles

The formal definition of a rake angle is "the

angle between the leading edge of a cutting tool and

a perpendicular to the surface being cut". Figure 10

shows the picture of the rake angle.

Rake angles come in two varieties, positive and

negative. If the leading edge of the cutting tools is

ahead of the perpendicular, the angle is, by definition,

negative. On the contrary, if the leading edge of the

cutting tools is behind the perpendicular, the angle is,




by definition, positive. When a negative rake angle

instrument is used, the material is cut by applying

downward pressure which creates a compressionwave ahead of the cutting tools. It's sort of like

spreading butter on a toast or scraping paint with a

plane cutting tools. It requires a lot of pressure to

keep the cutting tools in contact with the surface

being cut. This pressure and burnishing creates what

is known as a smear layer, which consists of a layerof crushed dentin, necrotic debris and bacteria that

gets burnished into the dentinal tubules and lateral

canals.

When a positive rake angle is used, on the other

hand, the material is cut by separating one molecule

of material from the work piece and creating a chipthat curls away from the edge of the cutting tools.

This cuts so readily that we have to make an effort to

keep the cutting tools from digging into the work-

piece. Compared to the negative rake angle, this is a

very efficient way of cutting.

Figure 10. Rake angle

6. Conclusion

In the simulation model, the most significant

influence on the forces is the chip thickness (Tc).

However, the dynamic radius caused by cutter

runout and tilt is another key point to affect chip

thickness. The effect of feed per flute on milling

forces is apparent from Figure 4 (a), (b), (c), and (d).

When feedrate is increased, the instantaneous chip

thickness is increased, and forces are increased.

Radial and axial depths of cut affect the width and

length of the contact area, respectively. That is,

when the radial and axial depth of cut are increased,

the contact area is increased, and the forces become

larger.

From Figure 11, it is obvious that when radial

depth of cut increases, the forces also increase.

Furthermore, the measured X forces change from

negative values to positive values when radial depth

of cut is changed from 25% to 75%.

-10

-8

-6

-4

-2

0

1 345 689 1033 1377 1721 2065 2409 2753

Samples

X F o r c e s

( l b )

(a) Radial D.O.C.= 33%

-6-5-4

-3-2-10

12

1 348 695 1042 1389 1736 2083 2430 2777

Samples

X F o r c e s ( l b )

(b) Radial D.O.C.= 50%

0

2

4

6

8

10

1 346 691 1036 1381 1726 2071 2416 2761Samples

X F o r c e s ( l b )

(c) Radial D.O.C.= 66%

Figure 11. Simulated X forces with respect todifferent radial depth of cut

Acknowledgment

The authors gratefully acknowledge the support

provided by the University of Kansas School of

Engineering, Department of Mechanical Engineering,

and Computer-Integrated Manufacturing Laboratory.

This work was also partially supported by a grant

from the National Institute of Standards and

Technology.

Reference

[1] Altintas Y., D. Montgomery and E. Budak,

"Dynamic Peripheral Milling of FlexibleStructures," Journal of Engineering for Industry,

Vol. 114, May, pp.137-145, (1992).

[2] Altintas Y., "Prediction of Cutting Forces and

Tool Breakage in Milling from Feed Drive

Current Measurements," Journal of Engineering

for Industry, Vol. 114, , pp.386-392, 2 November(199).




[3] Budak E. and Y. Altintas, "Prediction of Milling

Force Coefficients from Orthogonal Cutting

Data," Manufacturing Science and Engineering,PED-Vol. 64, , pp.453-460, ASME (19930.

[4] Fussell B. K. and K. Srinivasan, "An

Investigation of the End Milling Process Under

Varying Machining Conditions," Journal of

Engineering for Industry, Vol. 111 , pp.27-

36.,February (1989).[5] Fu H. J., R. E. DeVor and S. G. Kapoor, "A

Mechanistic Model for the Prediction of the

Force System in Force Milling Operations,"

Journal of Engineering for Industry, Vol. 106,

pp.81-88, February (1984).

[6] Fussell B. K. and K. Srinivasan, "On-LineIdentification of End Milling Process

Parameters", Journal of Engineering for Industry,

Vol. 111, pp.322-330, November (1989).

[7] Kline W.A., R.E. DeVor and J. R. Lindberg, "

The Prediction of Cutting Forces in End Milling

with Application to Cornering Cuts," Int. J. Mach.

Tool Des. Res., Vol. 22, No. 1 , pp. 7-22,(1982).

[8] Kline W.A. and R.E. DeVor, "The Effect of

Runout on Cutting Geometry and Forces in End

Milling," Int. J. Mach. Tool Des. Res., Vol. 23,

No. 2/3, pp. 123-140, (1983).

[9] Kline W. A., R. E. DeVor and I. A. Shareef,"The Prediction of Surface Accuracy in End

Milling", Journal of Engineering for Industry,

Vol. 104, pp.272-278, August (1982).

[10] Kolarits F.M. and W. R. DeVries, "A

Mechanistic Dynamic Model of End Milling forProcess Controller Simulation," Journal of

Engineering for Industry, Vol. 113, pp.176-183,

May (1991).

[11] Lauderbaugh L. K. and A. G. Ulsoy, "Dynamic

Modeling for Control of the Milling Process,"

Journal of Engineering for Industry, Vol. 110,

pp.367-375, November (1988).[12] Liang S.Y. and S.A. Perry, "In-Process

Compensation for Milling Cutter Runout via

Chip Load Manipulation", Journal of

Engineering for Industry, Vol. 116, pp. 153-159,

May (1994).

[13] Melkote S. N. and A.R. Thangaraj, "An

Enhanced End Milling Surface Texture Model

Including the Effects of Radial Rake and

Primary Relief Angles," Journal of

Engineering for Industry, Vol. 116, pp.166-174,

May (1994).

[14] Montgomery D. and Y. Altintas, "Mechanismof Cutting Force and Surface Generation in

Dynamic Milling," Journal of Engineering for

Industry, Vol. 113, pp.160-168, May (1991).

[15] Ramaraj T. C. and E. Eleftheriou, "Analysis of

the Mechanics of Machining With Tapered End

Milling Cutters," Journal of Engineering for

Industry, Vol. 116, pp. 398-404, August (1994).

[16] Smith S. and J. Tlusty, "An Overview of

Modeling and Simulation of the Milling

Process," Journal of Engineering for Industry,

Vol. 113, pp.169-175, May (1991).

[17] Sutherland J. W. and R.E. DeVor, "AnImproved Method for Cutting Force and

Surface Error Prediction in Flexible End

Milling Systems," Journal of Engineering for

Industry, Vol. 108, pp.269-279, November

(1986).

[18] Wang J.-J. Junz, S.Y. Liang and W. J. Book,

"Convolution Analysis of Milling Force

Pulsation," Journal of Engineering for Industry,

Vol. 116, pp.17-25, February (1994).

[19] Yen D. W. and P. K. Wright, "Adaptive

Control in Machining - A New Approach

Based on the Physical Constraints of ToolWear Mechanisms," Journal of Engineering for

Industry, Vol. 105, pp.31-38, February (1983).

Manuscript Received: Oct. 08, 1999

Revision Received: Apr. 10, 2000

And Accepted: Apr. 21, 2000

modeling of cutting forces in end milling operations

Documents