MATHEMATICAL MODELLING OF SURFACE ROUGHNESS IN
MILLING PROCESS
Dražen Bajić
Alen Belaić
Prof. dr. sc. D. Bajić, University of Split, FESB, R. Boškovića bb, 21000 Split
A. Belaić, University of Split, FESB, R. Boškovića bb, 21000 Split
Key words: up milling process, surface roughness, cutting parameters, mathematical
modelling
ABSTRACT
The roughness of technical surfaces presents one of the most important criterions
relating to the choice of machining process and cutting parameters in project processing. The
following article will therefore focus on researching results of the roughness of treated
surface depending on the cutting parameters characterized to the up milling process. In order
to find the most suitable empiric model to describe the depending character of the roughness
of treated surface the following elements have been taken into consideration: cutting speed,
the depth of cut and a feedrate. The experiment with the material 2011(AlCuPbBi) has been
conducted and the central composite design of the second degree has been applied. Given
results determined by regression analysis led to the empiric equation that is used for the
purpose of the calculation of the average arithmetic roughness.
1. INTRODUCTION
Production of technical surfaces of the machine components is realized either by the
method of chips forming machining or the one without metal removal. During the machining
and the usage of machine parts the given surfaces are exposed to the effect of the various
kinds of burdening. The most important ones are mechanical and chemical burdening that
result with dilapidation of parts and corrosion. Microscopically observed, technical surfaces
are not geometrical level surfaces with the ideal smoothness but rather rough level surfaces
characterized by series of uneven spots of different forms and disposition. The dimension of
the surface roughness can affect the following elements:
 the decrease of dynamic endurance,
 intensified friction and dilapidation of the tribo-burdened surfaces,
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Mathematical modelling of surface roughness in milling process
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 the decrease of overlap of a contraction link which effects the decrease of its carrying
capacity,
 speeding up the corrosion.
The roughness of the machined surface is seen through micro-geometrical
irregularities of the surface. The evaluation of the quality of machined surface is based on the
judgement of its roughness. Theoretical roughness depends exclusively on tools geometry
and applied process of machining whereas a real roughness appears as the result of
theoretical roughness though with bigger or lesser occasional roughness provoked by the
many factors. The surface roughness is influenced by the most important factors such as:





cutting parameters,
the work piece and tools' materials,
dynamic performance of machining system,
the coolant,
tool condition.
We can differ the roughness proceeded from direction of the main motion
(measurements are made in direction of machining) and the roughness came out of the
direction of feed motion (measurements are made vertically the traces of machining). The
first type of the roughness is characterized by the important activity of separation of scraps
whereas, beside the layer on the front surface, material and the geometry of tools, the cutting
speed plays the important role as well. However, the roughness dimension depends on
elastic deformation of tools, machine tool and the geometry of tools. The second type of
roughness is rather important for the surface quality and can be approximately measured on
the basis of geometry of the cutting part of tools and machining kinematics.
The main elements that determine the cutting parameters of chip forming machining
or rather the milling are as follows:
 the speed of metal removal (cutting speed) vc,
 depth of cut ap,
 feedrate vf.
2. THE AIM, METHODOLOGY AND CONDITIONS OF EXPERIMENT
Experiment aim is to define adequate mathematical model that is used to determine
the influence of independent factors and cutting data, at surface roughness. The following
independent factors are selected: cutting speed (vc), depth of cut (ap), and feedrate (vf).. For a
concrete case, that means determination of criteria:
– surface roughness: Ra = f(vc, ap, vf),
Experiments are performed in laboratory for machine tools of Faculty of electrical
engineering, mechanical engineering and naval architecture, University of Split, at universal
milling machine. The experiments are carried out by the HSS (DIN 841) milling tool as it is
shown at figure 1. For test piece material, a material 2011(AlCuPbBi) was selected, figure 2.
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Mathematical modelling of surface roughness in milling process
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Figure 1. – HSS milling tool (DIN 841)
60
50
5
5/45°
10
160
10
8
52
180
Figure 2. - Workpiece
The measurement of surface roughnes was made by «Surtronic 3» instrument. The
multifactor design of the second degree has been used to carry out this experiment. Actually,
in order to learn more about the maximum or minimum of the process or its function it is
necessary to approximate it by the polynomial of the second rather than the polynomial of the
first degree.
The selected values of the cutting parameters are the following:
– cutting speed:
vc,max = 78,539 [m/min]
vc,min = 40,055 [m/min]
– depth of cutt:
ap,max = 2 [mm]
ap,min = 1 [mm]
– feedrate:
vf,max = 190 [mm/min]
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Mathematical modelling of surface roughness in milling process
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vf,min = 90 [mm/min].
Central composite design with empiric polynomial model of the second degree is
taken:
k
k
k
i 0
1 i  j
i 1
y  b0   bi  xi   bij  xi  x j   bii  xi2
(1)
–- b0, bi, bij, bii – regression coefficient,
– x
– coded values of input parameters.
In order to get regression equation determined by polynomial of the second degree
using the statistics analysis, it is necessary to expand the design matrix with some other
physic factor values or rather to increase the number of experimental points which is to get
by rotatability character. Rotatability can be selected by an appropriate choice of coefficient,
marked by  value of which depends upon the number of the points of factorial design. For k
= 3 the given value of  = 1,682. The needed experimental points number, as far as the
design of the second degree is concerned, figure out the following:
N  2 k  n0  n  20
(2)
2k – the design number within the basic points
n0 – the repeated design number of the average level, n0 = 6
n – the design number on the central axes, n = 6
Adding the points to the central axes where xi = ± , and  = 1,682, the 3-factorial
design can be presented in Table 1.
Table 1. – Physic values and coded indexes of input factors
Coded values of input factors
x-i
x-i,min
x-i0
xi,max
x+i
-1,682
-1
0
+1
+1,682
x1 = vc [m/min]
21,991
40,055
60,475
78,539
95,033
x2 = ap [mm]
0,5
1
1,5
2
2,5
x3 = vf [mm/min]
32
90
135
190
255
Input factors
3. EXPERIMENTAL RESULTS AND STATISTICS ANALYSIS
Measured values of surface roughness, as the results of testing twenty experimental
points defined by experiment plan matrix, are shown in Table 2. The mentioned values of
surface roughness are input data for mathematical modeling of results, which was made by
multiple regression analysis and by using program package “Design Expert 6”.
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Table 2. – Given results of the surface roughness measurements
TEST
NUMBER
1
2
3
4
5
I6
I7
I8
I9
I10
I11
I12
I13
I14
I15
I16
I17
I18
I19
I20
INPUT
Measured
values
Calculated
values
vc [m/min
ap [mm]
vf mm/min
Ra m
Ra m
40,055
78,539
40,055
78,539
40,055
78,539
40,055
78,539
21,991
95,033
60,475
60,475
60,475
60,475
60,475
60,475
60,475
60,475
60,475
60,475
1
1
2
2
1
1
2
2
1,5
1,5
0,5
2,5
1,5
1,5
1,5
1,5
1,5
1,5
1,5
1,5
90
90
90
90
190
190
190
190
135
135
135
135
32
255
135
135
135
135
135
135
1,73
0,67
1,82
0,81
1,95
0,99
1,98
1,11
2,88
0,84
1,07
1,25
0,91
1,62
1,17
1,11
1,14
1,18
1,15
1,15
1,73
0,59
1,73
0,59
2,07
1,06
2,07
1,06
2,86
0,78
1,12
1,23
0,86
1,72
1,12
1,12
1,12
1,12
1,12
1,12
On Figure 3 comparison between predicted (calculated) and actual (measured)
values is presented.
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Mathematical modelling of surface roughness in milling process
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Figure 3.- Comparison between predicted (calculated) and actual (mesured) values for
surface roughness Ra
Final mathematical model of surface roughness Ra is obtained:
Ra  4,352  0,088  vc  4,674  104  vc2  7,796  106  v2f  3,285  105  vc  v f
(3)
with regression coefficient: r2 = 0,9904.
The analysis of obtained mathematical models and cutting data influence at surface
roughness has been made using diagrams shown in Figures 4, 5, 6.
Figure 4.- Surface roughness in dependence of cutting speed vc and feedrate vf
(ap= 1 [mm])
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Mathematical modelling of surface roughness in milling process
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Figure 5.- Surface roughness in dependence of cutting depth ap and cutting speed vc
(vf= 135 [mm/min])
Figure 6.- Surface roughness in dependence of cutting depth ap and feedrate vf
(vc= 65 [m/min])
4. CONCLUSION
Mathematical model presents quite well the performance of the average arithmetic
roughness. However, it can be used as well for the evaluation of the surface roughness value
in up milling process, whilst applying specific cutting parameters or as a useful model in
selection of appropriate cutting parameters in order to achieve a specific demanding
roughness.
Relating to the given equation it is worth to point out the following conclusions:

Regression coefficient r2 which determines quality and reliability of model is
0,9904, what means that 99,04 % of variability is caused by impact of
variables, meaning that mathematical model quite well describes correlation of
surface roughness Ra, to selected input factors vc, ap, vf .


The final value of surface roughness Ra is most dependent to cutting speed vc.
Feedrate vf also impacts surface roughness Ra. The correlation of feedrate to surface
2
roughness is presented in model through its quadratic value (v f ) and interaction with
cutting speed (vc  v f ) .

Impact of depth of cut ap is negligible by comparison with cutting speed vc and
feedrate vf. Therefore, depth of cut as well as its interactions with other two factors is
not included in final formula.
Relating to the Figures 4, 5 and 6., it is worth to point out the following conclusions:
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Mathematical modelling of surface roughness in milling process
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
Surface roughness Ra enhances with decrement of cutting speed vc and increment of
feedrate vf. Also, greater impact of cutting speed than feedrate on surface roughness
can be observed, figure 4.

Decrement of cutting speed vc and increment of depth of cut ap causes surface
roughness enhancement. Although impact of depth of cut by comparison with cutting
speed is almost negligible, it exists as result of vibrations and reduction of machine
stiffness, figure 5.

It is essential to point out that detailed study of diagrams gave following concludions:
At high feedrate values vf and low cutting speed values vc, enhancement of depth of
cut ap, results with improvement of surface roughness, while at higher cutting speed
values, enhancement of depth of cut results with enhancement of surface roughness.
5. LITERATURE
1
2
3
4
5
6
CEBALO, R; BAJIĆ D.; BILIĆ, B.: Mathematical modelling of cutting forces in the
longitudinal turning process, 10th International Scientific Conference on Production
Engineering CIM 2005, Lumbarda, 2005., str. I 31-I 40.
BILIĆ, B.; BAJIĆ, D.; VEŽA, I.: Optimization of cutting parameters regarding surface
roughness during longitudinal turning, 15th International DAAAM Symposium:
Intelligent Manufacturing & Automation: Globalisation – Technology – Men - Nature,
Vienna, 2004., str. 039-040.
BAJIĆ, D.: Doprinos poboljšanju obradivosti kod kratkohodnog honovanja, Ph.D.
Thesis, University of Zagreb, FSB, Zagreb 2000.
CEBALO, R.: Ovisnost dubine hrapavosti o srednjoj aritmetičkoj hrapavosti brušene
površine kod različitih postupaka brušenja, Strojarstvo 35(5,6) 231-235 (1993).
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New York, 1997.
CEBALO, R.: Prepoznavanje materijala i automatsko određivanje elemenata rezanja
kod tokarenja, Zbornik II, Suvremeni trendovi proizvodnog strojarstva, Zagreb, 1992.