Empirical Modeling of Cutting Forces in Ball End Milling ... · In case of geometric modeling, cutting edge of the ball-end mill was considered as a series of infinitesimal elements,

5th International & 26th All India Manufacturing Technology, Design and Research Conference (AIMTDR 2014) December 12th–14th, 2014, IIT

Guwahati, Assam, India

464-1

Empirical Modeling of Cutting Forces in Ball End Milling using

Experimental Design

VenkateswaraSarma M.N.M.1*,Manu. R.2

1*M. Tech. Scholar, NIT Calicut, 673601, [email protected] 2Associate Professor, NIT Calicut, 673601, [email protected]

Abstract

Precision parts with curved surfaces such as dies and molds are required in many manufacturing industries. Ball end

milling is one of the most common manufacturing processes for such parts. Force modeling of ball-end milling is

important for tool life estimation, chatter prediction, tool condition monitoring and to estimate the tool deflection

which affects the quality of the finished parts. This project presents an approach for modeling the cutting forces

acting on ball end mill in milling process. The steps used in developing the model are based on mechanistic

principles of metal cutting. Initially, the forces acting on the ball end mill are modeled based on the literature, in

which the empirical relationships were used to relate the cutting forces to the undeformed chip geometry.These

modeling equations governing the cutting forces are programmed in the MATLAB software. A series of slot milling

experiments are conducted using a ball end mill by varying the feed and depth of cut and the cutting forces acting on

the work piece are measured. An algorithm was developed, to calculate the empirical parameters, by using the

deviation between the average forces measured while doing experiments and the force values predicted by the

software program. Keywords:Force prediction, ball end mill, Algorithm

1. Introduction

The ball-end milling process is one of the most

widely used machining processes for the components

that are characterized with free-form surfaces like

dies, molds and various automotive components.

Machining parts like automotive and aerospace

components demands a high level of accuracy.

Cutting forces acting on the tool are the important

factors which governs chatter and surface quality

while machining.

In general, prediction of forces in flat end milling

(or any other conventional milling cutters) is not a

complex problem. But, in the case of ball end

milling, it is not easy, due to the complex geometry

of the ball end mill, which makes it difficult, either

for geometric modeling of the tool or simulating

using design software. Therefore different

approaches are being developed by researchers in this

area since many years. Force prediction gives the

manufacturer, a clear idea about the process being

done.

There are two important approacheswhich were

proposed in literature for estimating the cutting force

coefficients in milling process: orthogonal to oblique

cutting transformation approach and analytical fitting

of estimated cutting forces to the experimentally

measured ones.

In case of geometric modeling, cutting edge of

the ball-end mill was considered as a series of

infinitesimal elements, and the geometry of a cutting

edge element was analyzed to calculate the necessary

parameters for its oblique cutting process assuming

that each cutting edge was straight. Transformation

mechanics of orthogonal to oblique cutting define the

cutting force coefficients using the shear stress, shear

angle and friction angle. The coefficients are

determined from orthogonal turning experiments,

which are then transformed into oblique cutting edge

for prediction of cutting forces in helical end milling

forces.Specifically, for ball-end milling process,Yang

and Park (1990)developed a model using

orthogonalcutting data obtained from end turning

tests and then extended the model for flexible cutting

systems. Yucesan and Altintas(1994) evaluated the

varying rake face friction and pressure distribution

and the chip flowangle in peripheral milling in order

to provide accuratecutting force predictions.

Subsequent studies by Altintaset al. (1996) further

demonstrated that the milling force coefficients could

be determined from orthogonalcutting tests with

oblique cutting analysis and transformation.

In analytical modeling, initially, a force model is

developed based on the mechanistic principles of

metal cutting. The cutting forces are calculated on the

Empirical Modeling of Cutting Forces in Ball End Milling using Experimental Design

464-2

basis of the engaged cut geometry, the undeformed

chip thickness distribution along the cutting edges,

and the empirical relationships that relate the cutting

forces to the undeformed chip geometry. Now these

empirical relations should be estimated by real force

values obtained by conducting experiments.Feng and

Menq (1994) designed a cutting force model and

determined empirical cutting force coefficients using

numerical polynomial fit.Wan et al. (2006) did

numerical simulations for general end mills utilizing

specific data points that do not vary with respect to

the cutting force coefficients, as references. However,

these proposed methods have either some geometric

constraints in conducting the calibration test, which

are often different from the intended machining

geometries, or part of the analysis needs to be

conducted offline, which introduces delay and thus

hinder the use of these techniques to monitor and

control of the process.

In the present work, the force modeling was done

according to the mechanistic principles of metal

cutting, in which the forces acting on the ball end

mill are directly proportional to the undeformed chip

geometry. A set of model building experiments was

conducted and measured force values are used to

estimate the empirical coefficients using numerical

fitting procedure.

2.Cutting force model

The cutting force model was developed

based on Mechanistic principles of metal cutting in

which the forces are directly proportional to the

undeformed chip geometry. The forces acting on the

ball end mill are modeled based on the equations

from the paper given by Feng and Menq, in which the

empirical relations are used to relate the cutting

forces to the undeformed chip geometry. The

following simple formula shows the estimation of

cutting force with the size effect explicitly

considered:

� � ��(1)

where F is the principal cutting force responsible for

the total energy consumed, b is the width of cut, t is

the undeformed chip thickness, K is the cutting

mechanics parameter, and 1 > m > 0 for most

metallic materials. In the above expression, K

represents the condensed effects of all process

parameters except b and t, the undeformed chip

geometry parameters.

It is clear from equation (1) that a factor

essential to the prediction of cutting forces is the

undeformed chip geometry along the cutting edges

engaged with the work piece. If the cutting speed is

much larger than the table feed rate, the circular tooth

path can be used and a good approximation is

obtained for the instantaneous chip thickness

�� sin (2)

where f is the feed rate (mm/tooth) and θ indicates

the angular position of the cutting edge.

A complete representation of the

undeformed chip geometry along the cutting edges is

dependent on the geometric design of the ball end

milling cutter. Therefore, the angular position of a

differential element on the ith cutting edge of an n-

fluted cutter at a distance z from the free end can be

expressed as:

��, � � � �� tan � � �� 1 ��

� (3)

where is the angular position designated to tooth

number 1 (arbitrarily selected) at the free end (z = 0),

β is the helix angle, and R is the nominal radius of the

ball-end mill. Combining equations (2) and (3), the

undeformed chip thickness for the differential cutting

edge element is

��, � � � sin ��, �(4)

With equations (1) and (4), the general expression for

the elemental tangential cutting force �� and��may now be written as

�� , �!�"dz(5)

�� , �!�#dz (6)

where dz stands for the width of cut of the differential

cutting edge along the z direction.

In equations (5) and (6), ��and ��, which

basically characterize the local cutting mechanics of

the differential cutting edges at z, are approximated

by polynomial expressions. In order to evaluate the

associated empirical parameters with reasonable

efficiency, the following expressions were selected to

approximate�� and ��

�� $% & $' (�)* & $� (�

)*�& $+ (�

)*+

(7)



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�� ,% & ,' (�)* & ,� (�

)*�& ,+ (�

)*+

(8)

In the above expressions, z is non-

dimensionalized with respect to R and the two

domain representations of �� and �� are based

on the fact that the differential cutting edges on the

cylindrical part are geometrically the same and

exhibit the same cutting characteristics, while those

on the ball part are geometrically different from each

other, which results in varying cutting characteristics

along the axial direction.

To obtain the total forces acting on the ball-

end mill at any instant, the elemental tangential and

radial cutting forces are resolved into the external x,

y coordinate system and summed over all the

engaged differential cutting edges. The summation

(integration) is done numerically along the z-axis to

yield the instantaneous forces in the x, y directions:

�- � . /∑ �� , �!�" � cos ��, �! &��3'

4%

�� , �!�# � sin ��, �!5��(9)

�6 � . /∑ �� , �!�" sin ��, �! &��3'

4%

�� , �!�# � cos ��, �!5��(10)

In case of slot milling experiments, due to

the anti-symmetry of the cosine function to the same

axis, it can be neglected. It is then clear that slot

machining should be used in the model- building

experiments so that, in the numerical fitting

procedure, the tangential and radial model parameters

are decoupled and independent of each other

3. Programming in MATLAB

The modelling equations governing the

cutting forces were programmed in the MATLAB

software. Now by using this MATLAB program, we

can calculate the cutting forces, when the empirical

and geometric data is available.

4. Experimental Work

A series of model building experiments was

performed on Agni BMV45 TC24 4-axis vertical

machining centre. The carbide ball-end mills with 10

mm diameters, four right-handed flutes and 30° helix

angles were used.The ball-end mills were placed in

the collet-type tool holders and the work piece

isaluminium alloy 2024-T6 block. A total of 12 slot

cuts were carried out and the details of the cutting

conditions are shown in Table 1.

KISTLER dynamometer of Type 9257B was

used to measure the cutting forces while machining.

Itsmulti-channel charge amplifier Type 5070A for

multi component force measurement was used to

amplify the charge signal.

Table 1 Cutting conditions for model building

experiments

Depth

of

cut(mm)

Feed

rate(mm/min)

Spindle

speed

(rpm)

�- (N)

�6(N)

3 40 600 265.7 -178.6

4 40 600 479.6 -34.33

5 40 600 653.7 347.11

3 50 600 244.7 -

125.46

4 50 600 464.11 -42.5

5 50 600 818.7 431.7

3 60 800 219.7 -202.3

4 60 800 347.5 -38.3

5 60 800 735.6 387.5

3 70 800 256.4 -236.8

4 70 800 562.5 -44.6

5 70 800 859.4 453.7

5. Methodology

The MATLAB program for the model

equation of the cutting forces is used to estimate the

empirical parameters. An iterative numerical

algorithm is described in the Fig. 1. The algorithm

starts with an assumption that the empirical

coefficients obtained from the literature, also holds

good for the forces obtained from the slot milling

experiments. Those empirical coefficients are used to

find the forces with the geometric data available with

the slot milling experiments using the program. But

due to the difference in the tool, work piece material,

geometry of ball end mill and other physical

parameters, the predicted force values and

experimental values do not match each other. The


difference between the forces is used to approximate

the new empirical coefficients. Then, the algorithm

goes back, to estimate the model parameters again by

using the latest obtained empirical coefficients. This

procedure repeats itself until the estimates for the

model parameters stabilize numerically.

Fig. 1. An iterative algorithm to find empirical

parameters

The initial empirical parameters that are

taken from the literature review are as follows:

Table 2Iterative algorithm to find parameters in X

It. No.

1 5682 -7507

2 4529 -9439


difference between the forces is used to approximate

the new empirical coefficients. Then, the algorithm

goes back, to estimate the model parameters again by

using the latest obtained empirical coefficients. This

lf until the estimates for the

model parameters stabilize numerically.

Fig. 1. An iterative algorithm to find empirical

The initial empirical parameters that are

taken from the literature review are as follows:

(11)

5.1. Study of parameters

The approximation of the empirical

coefficients should be done in such a way that, a special

weightage should be given to each and every

coefficient. In order to find the role of each coefficient,

the study of parameters is done separately by varying

them (keeping the remaining parameters constant) and

observing the change in force values accordingly.

In order to give the weightage while

approximating the model parameters, slopes of the lines

representing the parameters individually are calculated.

The following are the slopes respectively.

Slope of line representing




An iterative algorithm is followed to find the

parameters as explained above. After each and every

iterative step, the deviation between the measured and

simulated forces was found and the model parameters

were modified according to the given weightage. The

algorithm stops when the deviation is less th

The following are the conditions for which the

parameters are estimated.

Depth of cut = 3mm

Feed = 40 mm/min

Spindle speed = 600 rpm

Measured force, = 265.7 N

Table 2Iterative algorithm to find parameters in X direction

Simulated

forces, (N) deviation

3416 225 738 472.4

189 -730 368 103

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(12)

The approximation of the empirical

coefficients should be done in such a way that, a special

weightage should be given to each and every

coefficient. In order to find the role of each coefficient,

the study of parameters is done separately by varying

(keeping the remaining parameters constant) and

observing the change in force values accordingly.

In order to give the weightage while

approximating the model parameters, slopes of the lines

representing the parameters individually are calculated.

lowing are the slopes respectively.

Slope of line representing =

= -

=

Slope of line representing =

An iterative algorithm is followed to find the

parameters as explained above. After each and every

iterative step, the deviation between the measured and

simulated forces was found and the model parameters

were modified according to the given weightage. The

n the deviation is less than 2%.


Percentage

deviation

178.26%

38.86%



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3 4077 -9883 -451 -938 287 22.6 8.52%

4 3821 -9975 -496 -983 269.9 4.93 1.82%

The slopes of the lines representing the effect

of model parameters are used to estimate the required

results. Initially, the parameters are taken from the

literature and the corresponding forces should be found.

There will be a deviation between the simulated and

measured values. Now, the parameters should be

changed in such a way that the deviation reduces after

each step. Let us assume that each coefficient

contributes equally for the deviation. Since there are

four parameters, divide the deviation into four equal

parts. Each parameter is modified according to their

slope respectively. Since it was assumed that each

coefficient contributes equally for the deviation, again

there will be deviation but less than the previous one.

After a few iterative steps, the deviation decreases and

simulated force values tend towards the measured

values.

The cut-off value for the deviation is 2%. The

above iterative algorithm was applied for 5 different

conditions and similar results have been obtained. Now

the average of all the model parameters at different

conditions gives the output model parameters.

$%= 3872

$'= -9872

$�= -514

$+= -992

5.2. Study of parameters78, 79, 7:, 7;

Finding the parameters in Y direction is also done

in the same way as explained above.

Slope of line representing ,% = '

�.=%

Slope of line representing ,' = '

'%.%>

Slope of line representing ,� = -'

�?.%+

Slope of line representing ,+ = '

@>.?=


parameters are estimated.

Depth of cut = 3mm

Feed = 40 mm/min

Spindle speed = 600 rpm

Measured force, �6 = -178.6 N

Table 3 Iterative algorithm to find parameters in Y direction

It. No. ,% ,' ,� ,+ Simulated

forces, �6(N) Deviation

Percentage

deviation

1 480 7055 -11953 5617 -400.44 222.14 124.53%

2 330 6497 -13397 2582 -220.7 41.7 23.38%

3 301 6392 -13668 2012 -186.7 8.4 4.7%

4 296 6371 -13772 1897 -180.51 2.21 1.23%

The average of all the model parameters at

different conditions gives the output model parameters.

,%= 282

,'= 6383

,�= -13705

,+= 1913

The final model parameters of the model

equation governing the cutting forces are as follows:

�� 3872 � 9872 (��* � 514 (�

�*�� 992 (�

�*+

(13)


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�� 282 & 6383 (��* � 13705 (�

�*�&

1913 (��*

+ (14)

6. Model validation

The above obtained model parameters

were used to estimate the forces and then compared

with the measured values. The verification experiments

indicate that the magnitudes of the predicted forces

correspond well with the actual test results.

Table 4 Validation of results in X direction

Sl. No. Depth of

cut(mm)

Feed

rate(mm/min)

Spindle

speed(rpm)

Simulated force,

�-(N)

Measured

Force, �- (N)

Deviation

percentage

1 3 60 800 224.4 219.7 2.13%

2 4 70 800 546.8 562.6 2.8%

3 5 50 600 800.6 818.7 2.22%

TABLE 5 Validation of results in Y direction

Sl. No. Depth of

cut(mm)

Feed

rate(mm/min)

Spindle

speed(rpm)

Simulated force,

�6(N)

Measured

Force, �6 (N)

Deviation

percentage

1 3 60 800 -195.8 -202.3 3.21%

2 4 70 800 -43.3 -44.6 2.91%

3 5 50 600 422.6 431.7 2.11%

7. Conclusion

The above model could successfully

predict the forces developed during ball end milling

operations. This was validated using slot milling

operations at different cutting conditions. This model

can be extended to develop a more accurate and

practical model for the ball-end milling process so as to

provide guidance in selecting machining conditions in

free form surface machining.

References

Azeem, A., Feng, H.Y., Wang, L. (2004), Simplified

and efficient calibration of a mechanistic cutting

force model for ball-end milling, International

Journalof Machine Tool and Manufacture, Vol. 44,

pp. 291–298.

Dhupia, J., Girsang, I. (2012), Correlation-based

estimation of cutting force coefficients for ball-end

milling, Machining Science and Technology: An

International Journal, Vol. 16, No. 2, pp. 287-303.

Feng, H.Y., Menq, C.H. (1994), The prediction of

cutting forces in the ball-end milling process—I,

Model formulation and model building procedure,

International Journal Machine Tools Manufacturing,

Vol 34, No. 5, pp. 697–710.

Lee, P., Altintas, Y.(1996), Prediction of ball-end

milling forces from orthogonal cutting data,

International Journal of Machine Tools and

Manufacture, Vol. 36, No.9, pp. 1059–1072.

Wan, M., Zhang, W.H., Tan, G., Qin, G.H. (2007), An

in-depth analysis of the synchronization between the

measured and predicted cutting forces for

developinginstantaneous milling force model,

International Journal of Machine Tools &

Manufacture, Vol. 47, pp. 2018–2030.

Yang, M., Park, H. (1991), The prediction of cutting

force in ball-end milling, International Journal of

Machine Tools and Manufacture , Vol. 31, No. 1, pp.

45–54.

Yucesan, G., Altintas, Y., (1994), Improved modeling

of cutting force coefficients in peripheral milling,

International Journal of Machine Tools and

Manufacture, Vol. 34, No. 4, pp. 473-487

Yuwen, S., Ren, F., Guo, D., Jia, Z., (2009),

Estimation and experimental validation of cutting

forces in ball-end milling of sculptured surfaces,

International Journal of Machine Tool and

Manufacture, Vol. 49, pp. 1238–1244.

Documents

Empirical Modeling of Cutting Forces in Ball End Milling ... · In case of geometric modeling, cutting edge of the ball-end mill was considered as a series of infinitesimal elements,