34th INTERNATIONAL CONFERENCE ON
PRODUCTION ENGINEERING
28. - 30. September 2011, Niš, Serbia
University of Niš, Faculty of Mechanical Engineering
MATHEMATICAL MODELLING OF CUTTING FORCE AS THE MOST RELIABLE
INFORMATION BEARER ON CUTTING TOOLS WEARING PHENOMENON
Obrad SPAIĆ, Zdravko KRIVOKAPIĆ, Rade IVANKOVIĆ
Production and Management Faculty in Trebinje, East Sarajevo University, Trg palih boraca 1, Trebinje, Republika
Srpska, Bosna i Hercegovina
sobrad1@teol.net, zdravkok@ac.me, rade.ivankovic@gmail.com
Abstract: Being one of their prominent exploitative characteristics, cutting tools durability depends on
the character, intensity and the speed of wearing. Identification of tool wearing is of great significance
for the purpose of avoiding sooner or later replacement of tools. The parameters of tool wearing can be
measured by out-process and in-process-measuring systems. Given the extremely limiting role of the
former in modern production lines, development of the latter (the indirect measuring systems) has gained
prominence. The basis of indirect measuring systems comprises a set of various signals originating from
the units of the system under treatment which stand in certain correlations with the wearing parameters.
The paper presents mathematical models of axial force designed on the basis of experimental research in
drilling tempered steel by twist drills made of high-speed steel manufactured by powder metallurgy.
Key words: tool, durability, wear, cutting force, mathematical model
1. INTRODUCTION
2. AXIAL CUTTING FORCE IN DRILLING
Identification of cutting elements wear is of high practical
importance because, apart from allowing for timely
replacement of tools, it also allows for management of
wearing processes as well as for automation of treatment
and technological processes. Production lines, in
particular those of mass and large scale automated
production benefit from timely replacement of cutting
tools as it eliminates the low quality of final products and
reduces production costs arising from sooner and later
replacement of tools.
As in modern production lines the out-process methods of
measuring tool wear have become a significantly limiting
factor, development of online process measuring systems
are gaining prominence. The process methods most
frequently applied are the indirect ones whose basis
comprises a set of various signals originating from the
units of the system under treatment which stand in certain
correlations with the wear parameters. The information
bearers (signals) of tools wear in cutting process that are
most often used by researchers are cutting force and
resistance. Thus, J. Sheikh-Ahmad and R. Yadav [1]
designed a force model at milling composite materials by
applying regression analysis. B. Lotfi, Z. W. Zhong and
L. P. Khoo [2] designed a model to predict cutting force
at milling dependent on tool orbit. J. T. Lin, D.
Bhattacharyya and V. Kecman [3] showed that measuring
cutting force enables for tool wear to be monitored
without interruption of cutting process. Based on the
comparative analysis of the assessment of wear of a 8 mm
diameter twist drill, C. Sanjay, M. L. Neema and C. W.
Chin [4] showed that modified regression equations could
be used to asses the value of tool wear.
Axial cutting force in drilling (the auxiliary movement
resistance), F3, along with other conditions unaltered, is in
the function of cutting regime (the twist drill nominal
diameter, the spindle speed and feed):
F  f (D, n,s)
(1).
By way of experimental-analytical method, i.e. the theory
of experiment planning and the theory of regression
analysis, the cutting force can be expressed in the form of
a degree function:
F  CF  Db1F  n b2F  sb3F
(2),
where:
CF, b1F, b2F, b3F – are the constants dependent on the type
of material,
D [mm] – is the drill nominal diameter,
n [rev/min] – is the spindle speed, and
s [mm/rev] – is the feed.
3. EXPERIMENT PLANNING
With the aim to design a mathematical model of axial
force as the information bearer on the wear phenomenon,
the paper contains experimental research on the basis of
which the constants CF, b1F, b2F, b3F were determined
within the assumed mathematical model (2). Given that
the axial force is in the function of three parameters (D, n,
s), the experiment was conducted in line with the
complete three-factor orthogonal first-order plan, i.e. the
Box-Wilson’s plan, repeating the experiment four times in
the central plan point (n0=4) as illustrated in Table 1.
Table 1. The three-factor plan matrix
Experimental
points
1
2
3
4
5
6
7
8
9
10
11
12
Coded values
x0
x1
x2
x3
x1x2
x1x3
x2x3
x1x2x3
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
-1
+1
-1
+1
-1
+1
-1
+1
0
0
0
0
-1
-1
+1
+1
-1
-1
+1
+1
0
0
0
0
-1
-1
-1
-1
+1
+1
+1
+1
0
0
0
0
+1
-1
-1
+1
+1
-1
-1
+1
0
0
0
0
+1
-1
+1
-1
-1
+1
-1
+1
0
0
0
0
+1
+1
-1
-1
-1
-1
+1
+1
0
0
0
0
-1
+1
+1
-1
+1
-1
-1
+1
0
0
0
0
3.1. Experiment conditions
The experiment was conducted by means of twist drills
(TD) DIN 338, nominal diameter Ø6.0; Ø7.75 and Ø10.0
mm, made of high-speed steel with 8% of Co,
manufactured by powder metallurgy, used for drilling
blind hole in tubes made of chrome-molybdenum alloy
steel for enhancement, Č.4732, thermally treated to 43-45
HRc hardness.
The chemical composition, thermal treatment conditions
and microstructure of the TD steel and test tubes are
shown in References [5].
Construction of twist drills followed the recommendations
in References pertaining to drilling hard treatable
materials (tempered steel) as well as previous experience.
The drills were manufactured by grinding technology.
The geometrical elements of TD are shown in References
[5].
The experiment was conducted using test tubes, Ø60 mm
in diameter, with the thickness adjusted to the blind hole
drilling depth 1=3xd.
Test tubes’ hardness was evenly distributed along
longitudinal and cross cut and within the prescribed
limits.
Cutting regimes were defined in line with the
recommendations stated in References [5], adhering to the
interval limits of variations of influential factors
( n sr2  n min  n max , and s sr2  s min  s max ) and are shown in the
three-factor plan matrix (Table 1).
The experiment was conducted in the laboratory of the
Faculty of Mechanical Engineering in Podgorica,
University of Montenegro, on the universal milling
machine Typ: FGU-32. Integrated with the milling
machine was the equipment for measuring axial force and
torque manufactured by KISTLER.
During the experiment, the 8% solution of Teolin H/VR
in the quantity of 1 l/min was used for cooling and
lubrication.
3.2. Axial force measuring
Axial force measuring was conducted by means of a
three-component
dynamometer
manufactured
by
“Kistler“, TYP 8152B2, with the measurement range
Real values
The output
s
vector
d
n
[F]
[mm] [rev/min] [mm/rev] [mm/min]
6.0
10.0
6.0
10.0
6.00
10.0
6.0
10.0
7.75
7.75
7.75
7.75
250
250
500
500
250
250
500
500
355
355
355
355
0.027
0.027
0.027
0.027
0.107
0.107
0.107
0.107
0.053
0.053
0.053
0.053
6.67
6.67
13.33
13.33
26.67
26.67
53.33
53.33
18.67
18.67
18.67
18.67
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
F12
from 100 to 900 kHZ, integrated with the universal
milling machine and Global Lab software.
For the data acquisition during the experiment, the
KISTLER high-frequency amplifier, type AE-Piezotron
Coupler 5125B was used while a two-channel DAQ
Scope PCI-5102 was used as an AD convertor. The
acquired signals were processed by means of a virtual
instrument, aided by Global Lab software. The data
acquisition schema is illustrated in Figure 1.
Figure 1. Data acquisition schema during the experiment
4. EXPERIMENT RESULTS
Measurement of axial force was conducted in line with
the stated Plan matrix in five measuring points. The first
measurement was conducted during drilling of a blind
hole, l=3d, with sharp twist drills while the fifth (the last)
measurement was conducted subsequent to the achieved
lengths of drilling (in mm) during which the twist drills
wear reached the following maximally allowed (defined
in advance) values:
for TD Ø6.0 mm – 0.25 mm
for TD Ø7.75 mm – 0.30 mm
for TD Ø10.0 mm – 0.35 mm
The mean value of the wear band width of the back
surfaces was taken to represent the maximum wear value
(B ≈ 0.04D), at the edge of regular area which is at
0.025mm distance from outer fibres (Figure 2).
At different cutting regimes (the nominal diameter, the
spindle speed and the feed), TD reached the maximally
Assessment of the significance of the first-order model
was done by means of application of F-criteria, for the
adopted level of significance q=0.05, according to the
pattern:
Frj  N
Figure 2. Twist drill wear
allowed wear value at different drilling lengths. The
measured axial force value at maximal wear is shown in
Table 2.
Table 2. Axial force values at maximal TD wear
Exp.
points
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
b 2j
S2E
,  j  1, 2,3
(7).
Dispersion of experiment results in the multi-factor space
is as follows:
S2E 
1
fE
n0
å  y0i  y0 
2
(8).
i 0
S2E = 0.007 ,
D
n
s
lmax
Fmax
[mm] [rev/min] [mm/rev] [mm]
[N]
6.0
250
0.027
1265.40 956.50
10.0
250
0.027
2700.00 594.50
6.0
500
0.027
1533.00 1090.50
10.0
500
0.027
2340.00 1139.00
6.0
250
0.107
999.00 955.50
10.0
250
0.107
4800.00 638.50
6.0
500
0.107
1800.00 1925.50
10.0
500
0.107
756.00 1579.70
7.75
355
0.053
2569.13 783.30
7.75
355
0.053
2336.63 682.90
7.75
355
0.053
2569.13 655.50
7.75
355
0.053
2801.63 754.80
where:
fE = n0-1 = 3 – is the degree of freedom of experimental
error.
The degrees of freedom of parameters bj, (j = 0, 1, 2, 3)
are:
fbj = 1, which results in the tabled value of the dispersion
relation Fri,1-q/1, 3 = 10.10 [7].
The parameters of the model and assesments of
significance are listed in Table 3, which shows that all
parameters of the first-order model (b0, b1, b2 i b3) are
significant.
Table 3. Model parameters
Model parameters Disper. relations (Fri) Assesments
5. MATHEMATICAL
AXIAL FORCE
MODELLING
OF
The pattern of axial cutting force (2) subsequent to
linearisation can be expressed in the following form:
y  b0  b1x1  b2 x 2  b3 x3
(3),
where:
y = lnF, b0 = lnCF, x1 =ln(D), x2 =ln(n) and x3 =ln(s).
The orthogonal first-order plan with constant members
can be applied to the above pattern with coding performed
by means of the equations of transformation:
X1  2
X2  2
X3  2
ln  D   ln  Dmax 
ln  Dmax   ln  Dmin 
ln  n   ln  n max 
ln  n max   ln  n min 
ln  s   ln  s max 
ln  s max   ln  s min 
 1,
1
76831.617
16.813
19.112
102.457
0.575
12.214
9.484
1.724
sign.
sign.
sign.
sign.
not sign.
sign.
not sign.
not sign.
ŷ  6.820  0.1215x1  0.121x 2  0612x 3
(9).
5.358  D0.822  s0.433
(5),
N
(6).
(10).
n 0.372
(4).
while parameters bj are determined on the basis of results
of N = 2k points arranged along the vertexes of hypercube
according to the pattern:
1
å x ji  yi ,  j  1, 2,3
N i 0
Fb0
Fb1
Fb2
Fb3
Fb12
Fb13
Fb23
Fb123
On the basis of listed parameters the empirical model was
deduced of axial force at maximal TD wear:
F
By applying regression analysis, the parameter of the
model b0 is determined on the basis of results of all N = 2k
+ n0 plan points, in line with the pattern [6]:
bj 
6.820
0.121
-0.129
0.298
-0.021
0.103
0.091
-0.039
Revisiting the original coordinates by means of the
equations of transformation (4) provided for deduction of
a concrete empirical axial force model:
 1 , and
1 N
b0  å x 0i  yi
N i 0
b0
b1
b2
b3
b12
b13
b23
b123
Testing of adequacy of the defined model was done
according to Fisher’s criterium [6]:
FrLF 
S2LF
(11).
S2E
FrLF  15.061.
Dispersion of experiment results in the multi-factor space
is as follows:
S2LF

n0
k
1 N  2
2

y

N
b


 u  i  y0u  y0
f LF  u 1 
i 0
u 1







2



(12).
S2LF  0.1048 .
6. CONCLUSION
The degree of freedom of model adequacy is:
fLF=N-k-1-(n0-1)=5, thus entailing the tabled value of
dispersion relation of Ft(5%;5;3) = 9.0 [7].
Given the FrLF > Ft(5%;5;3) the mathematical model fails to
describe the correctly observed function of response,
which implies existence of effects of mutual action among
the model parameters.
Three-factor orthogonal first-order plans allow for
assessment of basic effects as well as the effects of mutual
action of the first and second order on the empirical
model of the response function. Should the complete
three-factor first-order model be applied, the pattern to
describe the (axial force) response function becomes nonlinear and can be expressed in the form of the equation:
The mathematical models designed by means of
orthogonal first-order plan and regression analysis fail to
properly describe the response function, i.e., axial cutting
force within the boundaries of the covered multi-factor
space due to mutual action of the model parameters, the
squared elements action effects, as well as the action of a
series of parameters whose source, nature and span of
action are unknown.
This points to the fact that highly complex processes that
take place in the zone of drilling tempered steel and which
are conditioned by actions of numerous influential and
mutually collinear factors, cause difficulties for
mathematical models to be applied so as to describe the
behaviour of mechanical, thermo-dynamical, tribological,
chemical and other phenomena in the cutting zone.
y = b0 +b1x1+b2x2+b3x3+b12x1x2+b13x1x3+
b23 x2x3+b123 x1x2x3
(13).
On the basis of values of the model parameters
corresponding to mutual action effects (see Table 4) and
taking into account the assessment of significance, while
also revisting the original coordinates by means of the
equations of transformation (4), the following non-linear
empirical model of axial force is deduced:
lnF = 6.82+2.187∙ln(D)+0737∙ln(n)-2.994∙ln(s)+
0.586ln(D)∙ln(s)+0.38∙ln(n)∙ln(s)-7.536
(14).
Given that the dispersion relation value of the parameter
b13 (9.484) is approximate to the tabled value (10.10), its
influence has been included in the model. The modelled
values of axial force, as well as the relative error in
relation to the experimental results are shown in Table 4.
Table 4. Modelled values of axial force
Exp. Modelled Error Exp. Modelled Error
points values
%
points values
%
1
830.47 -13.19
5
1024.26
-6.07
2
860.2
-9.97
6
1602.63 -16.75
3
535.15
-9.98
7
948.21 -16.75
4
554.3 -13.19
8
1483.64
-6.08
9-12
907.81
26.24
If we compare the modelled and experimental values of
axial force in the experimental points, we can see that
maximal deviation of experimental points from modulled
surface is 16.75%, thus providing for the model (14) to be
considered a mathematical interpretation of the goal
function. However, in the central plan point the deviation
of modulled results from the experimental ones is
26.24%, which means that mathematical model fails to
properly descibe the response function within the
boundaries of the covered multi-factor space. Therefore,
the null hypothesis that the effects of squared elements in
the model equal zero must be jettisoned.
For the purpose of designing a mathematical model in
order to properly decribe axial force within the boundaries
of the covered multi-factor space it is necessary to apply
the second-order polynomial model which necessitates
additional amount of information on the diffusion system,
i.e. additional number of experiments.
REFERENCES
[1] Sheikh-Ahmad J., Yadav R. (2008) Model for
predicting cutting forces in machining CFRP,
International Journal of Materials and Product
Technology, Vol. 32, Number 2-3, 152 – 167
[2] Lotfi B., Zhong Z. W., Khoo L. P. (2009) Prediction
of cutting forces along Pythagorean-hodograph
curves, The International Journal of Advanced
Manufacturing Technology, Vol. 43, Numbers 9-10,
872-882
[3] Lin J. T., Bhattacharyya D., Kecman V. (2003)
Multiple regression and neural networks analyses in
composites machining, Composites Science and
Technology, Vol. 63, 539–548
[4] Sanjay C., Neema M. L., Chin C. W. (2005)
Modelingof tool wear in drilling by statistical
analysis and artificial neural network, Journal of
Materials Processing Technology, Vol. 170, 494–500
[5] Spaić O. (2006) Uporedna analiza habanja zavojnih
burgija od brzoreznog čelika proizvedenog
konvencionalnom metalurgijom i metalurgijom
praha, MSc thesis, University of East Sarajevo,
Faculty of Production and Management Trebinje
[6] Stanić J. (1986) Metod inženjerskih mjerenja, Faculty
of Mechanical Engineering Beograd
[7] Laković R., Nikolić B. (1999) Primijenjena statistika
2. dio – eksperiment, University of Montenegro,
Faculty of Electrical Engineering, Podgorica
CORRESPONDENCE
Obrad SPAIĆ, PhD, Assoc. prof., University of East
Sarajevo, Faculty of Production and Management,
Trebinje, Trg palih boraca br. 1, Trebinje,
sobrad1@teol.net
Zdravko KRIVOKAPIĆ, PhD, Prof., University of
Montenegro, Faculty of Mechanical Engineering,
Podgorica, Džordža Vašingtona bb, Podgorica,
zdravkok@ac.me
Rade IVANKOVIĆ, PhD., Prof., University of East
Sarajevo, Faculty of Production and Management
Trebinje, Trg palih boraca br. 1, Trebinje,
rade.ivankovic@gmail.com