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HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPRÉM 
Vol. 39(2) pp. 189-193 (2011) 

METHOD FOR DETERMINATION OF THE EXPECTED ROUGHNESS OF  
CUT SURFACES 

C. FELHO , J. KUNDRAK 

University of Miskolc, Department of Production Engineering, H-3515 Miskolc, Egyetemvaros, HUNGARY 
E-mail: csaba.felho@uni-miskolc.hu 

 

Nowadays the manufacturing companies create stricter and stricter demands on quality requirements of surfaces machined 
by cutting. The fulfilment of these requirements is often not an easy task, and in addition technologists should deal with 
the economic aspects too. Surface roughness indexes are important parameters in characterization of the quality of parts, 
especially in case of engaging and sliding one another machine elements influences remarkably their tribological properties 
and lifespan. Therefore it is indispensably important to completely satisfy the surface roughness requirements. It can 
mean a great help, if the prospective value of roughness indexes can be predicted in advance for the actual machining 
environment, since the selection of the optimal tool-geometrical and technological parameters become possible. Such a 
method will be introduced in the paper, by the help of which theoretical values of surface roughness can be determined 
for single point cutting by the help of an analytical model. A general mathematical model is introduced, by the help of 
which edge geometry of every realistically evolving tool can be described. Theoretical values of the relevant roughness 
indexes used in practice can be calculated by the application of this model. The connection with the real roughness data 
gained by experiments can be written down after this, hereby the surface roughness can be predicted, and thus the 
planning of technology can be faster and more efficient.  

Keywords: surface roughness, theoretical roughness. 

Introduction 

The accelerated technological evolution, the higher and 
higher quality requirements demanded for machine parts 
are provoked increased claims also against manufacturing 
technologies. This fact is especially prevalent in the 
automotive industry. More and more tasks are automated, 
and increasing number of characteristics is described by 
theoretical and computer models. One important factor 
for the characterization of surface quality is the 
microgeometry of the surface profile. The microgeometry 
can be quantified by practically selected roughness 
indexes. The determination of theoretical values of 
surface roughness in different metal cutting processes 
has been in focus since about a half century. A variety of 
theories were created, and the problem was examined by 
many kinds of aspects. The determination of theoretical 
roughness is important for technologists, since the 
expectable value of real roughness can be predicted for 
a given procedure, and thus it can provide help for the 
computer aided planning of manufacturing processes. 

Benardos and Vosniakos [1] have made a good review 
about the research works created in this field. They have 
classified the papers to four categories: analitical, based 
on designed experiment, experimental and AI-based 
works. Based on their work, main research groups dealing 
with surface roughness are shown In Table 1 As it can be 
seen from the table, there are several different solutions 
to this problem, and each has its own advantages and 
drawbacks [2]. The set of parameters that are thought by 
Benardos and Vosniakos in [1] to influence surface 
roughness and thus have been investigated by researchers 
is diagrammatically represented in Fig. 1. 

A theoretical model is introduced in the article by 
the help of which the theoretical value of roughness 
indexes used in practice can be determined for every 
realistically feasible cutting tool design. Through the 
examination of theoretical roughness it is assumed, that 
the imprint (negative) of the part of the tool which is 
near to the nose will remain in the machined surface, 
and that will form the surface roughness profile, and this 
geometry can be calculated from the tool-geometrical 
and technological parameters. 

 



 

 

190

Surface Roughness

Cutting Tool Properties

Machining Parameters

Workpiece Properties

Cutting Phenomena

Workpiece length Workpiece hardness

Workpiece daimeter

Cutting force variation

Friction in the cutting zone

Chip formation

Vibrations

Accelerat ions

Nose radius

Runout errors

Tool material

Tool shape

Process kinematics

Cooling fluid

Stepover
Depth of cut

Tool angle
Feed rate

Cutting speed

 
Figure 1: Fishbone diagram with the parameters that affect surface roughness [1] 

 

Table 1: Main research groups dealing with modelling of surface roughness according to Benardos&Vosniakos [1] 

Modelling 
procedures and 
techniques 

Currently active 
research groups 

Summary of the present 
status 

Main disadvantages of the 
method 

Analytical models 
or computer 
algorithms 

Grzesik; Lin and Chan; 
Baek et al.; Chen et al.; 
Ehmann and Hong; Kim 
and Chu; Lee et al. 

● These ones simulate the 
cutting process in terms of 
kinematics and cutting tool 
properties. 

● Often includes other 
parameters (e.g.. vibration) 

● The obtained results are 
fairly good. 

A lot of other factors that 
contribute to the roughness 
formation mechanism are not 
considered (wear and deflection 
of the cutting tool or certain 
thermal phenomena). 

Based on  
experiments 

Abouelatta and Madl; 
Ghani and Choudhury; 
Jang et al.; Beggan et al.; 
Dhar et al.; Munoz-
Escalona and Cassier; 
Thielle and Melkote; 
Baptista and Antune 
Simoes; Coker and Shin; 
Diniz and Filho; Heisel 

● This is the most conventional 
approach adopted. 

● It is not difficult to 
implement and depending on 
the level of understanding of 
the participating phenomena, 
it can produce very good 
results. 

The obtained conclusions have 
little or no general applicability.

Based on designed 
experiments 

Davim; Choudhury and El-
Baradie; Feng and Wang; 
Kopac and Bahor; Thomas 
and Beauchamp; Alauddin 
et al.; Fuh and Wu 

● The response surface 
methodology (RSM) and the 
Taguchi techniques for 
design of experiments (DoE) 
are the mostly used methods.

The obtained relation has no 
general applicability. 

Based on artificial 
intelligence (AI) 

Azouzi and Guillot; 
Varghese and 
Radhakrishnan; Chien and 
Chou; Suresh et al.; Lee et 
al.; Li et al.; Oxley; Lin et 
al.; Matsumura et al.; 
Benardos and Vosniakos; 
Tsai et al.; Lou and Chen; 
Ho et al. 

● Generally the artificial neural 
network (ANN) models or 
genetic algorithms (GA) are 
used. 

● These can produce very good 
results and simultaneously 
offer the possibility for on-
line monitoring and/or 
control of the process. 

● It is very demanding on 
computational power. 

● There is no guarantee for 
their resulting performance 
in an application. 

 



 

 

191

Model for the calculation of theoretical roughness 
indexes in single point cutting 

A new mathematical model was developed for the 
analytical calculation of theoretical roughness indexes 
in turning operations [3]. The examined roughness 
parameters were Rmax, Rz, Ra and the tp bearing ratio, 
however, one great advantage of this model is, that it is 
possible to calculate virtually all roughness parameters 
used in practice. The calculation method is introduced 
for turning cylindrical surfaces, but the model can be 
easily transformed for cutting other types of surfaces 
and for other processes, e.g. to face milling. 

As it was observed from the literature overview, all 
of these analytical models were created with several 
simplifications, since there are too much factors affecting 
the real roughness profile [4]. The most important 
simplifications which were applied to our model are: 
● the workpiece material is un-deformable in the 

machined surface; 
● the cutting system is fully rigid; 
● the cutting edge of the tool is a defined geometrical line. 

 

 
Figure 2: The overview of the applied method 

 
The effect of these parameters can be built into the 

model by modifier factors. Series of experiments has to 
be performed for this, and the effect of the actual 
parameter can be measured and quantified. An overview 
of the applied calculation method is depicted in Fig. 2. 

The essence of this calculation method is to create the 
general mathematical model of the single point cutting 
tools which have all of the possible edge sections a 
practically feasible tool can have. By putting this 
imaginary tool to an x-y coordinate system with the 
intersection point of the main and the auxiliary edge 
(the tool peak point) in the origin, the respective edge 
sections of the tool can be described by coordinates and 
emplacement angles (Fig. 3).  

The general cutter profile can be described with the 
following equation: 

 ( )1 1 1L f , , , , ,′ ′′ ′ ′′= l l l l l l  (1) 
where: 

ℓ, ℓ´, ℓ˝ – sections of the main edge; 
ℓ1, ℓ1´, ℓ1˝ – sections of the auxiliary edge.  
 

x

y

1

2

3
4 5

6

7

R

"

'

1

1'

1"

κ
r2

κ
r1

κr
1´

κr
2´

x3 x5
x2 x6

x1=h1 x7=h2

y1
 =

 h
3

y7
 =

 h
4

y6

y5y3

y2

 
Figure 3: The general tool in the coordinate system 
 
Edge sections of the main edge can be described with 

the following equations: 

 

2 2

4 3

r 1 3 2

r 2 2 2 1

R R x ; x x x

tg x b ; x x x

tg x b ; x x x

= − − ≤ <

′ = κ ⋅ + ≤ <

′′ = κ ⋅ + ≤ <

l

l

l

 (2) 

Edge sections of the auxiliary edge can be written as: 

 

2 2

1 5 4

1 r 1 1 6 5

1 r 2 3 7 6

R R x x x x

tg ' x b ; x x x

tg ' x b ; x x x

;
κ

κ

= − − ≤ <

′ = + ≤ <

′′ = + ≤ <

l

l

l

 (3) 

The geometry of the actual tool can be deducted 
from this general model by the appropriate substitution 
of function parameters.  

The roughness profile can be calculated by making a 
profile shift in the feed direction by the feed rate value 
(Fig. 4). As it can be seen from the picture, the auxiliary 
edge sections of the forwarded tool (yaux1, yaux2 and yaux3, 
depending on the feed value) and the main edges of the 
tool in the origin (ymain1, ymain2, ymain3) are forming the 



 

 

192

roughness profile. The presented model assumed that a 
cylindrical surface will be generated. In case of conical 
turning, the profile shift has to follow the taper edge.  

 

x

y

f

1'

2'

3'

4'
5'

6'

7'

P

1

2

3

4
5

6

7

ymain1

ymain2

ymain3

yaux1

yaux2

yaux3

  
Figure 4: Typical points of the idealized roughness 

profile, main and auxiliary edges 
 

The next step is to calculate the intersection point P 
of the main and auxiliary edges. This can be performed 
by calculating the x,y coordinates, where the equations 
of the respective edge sections have equal values: 
yaux n = ymain m. The yp ordinate of this P point gives the 
maximal height of roughness profile (Rmax) if the depth 
of cut (ap) is greater than the ordinate of this point, or if 
ap > yp, then Rmax = ap. In conical turning, the distance 
of the Rz value (ten-point height of the profile) is equal 
to this calculated Rmax value in the theoretical profile, 
since there are no deviations. The calculation of the mean 
line of the profile, which has equal areas below and above 
is needed for the determination of Ra and tp values. This 
can be done by the least-square approximation method. 
The Ra value can be calculated from the generalized 
equation 

 
( )

l

0

a
1

R y x dx
l

= ⋅∫  (4) 

The material ratio tp can be calculated from the following 
equation: 

 

( )pm pf
p

x x
t 100 [%]

f

−
= ⋅  (5) 

where: 
xpm, xpf – intersection points of the profile sectioning 
level p with the main and the auxiliary edges 
respectively. 
 
The actual surface micro-profile is significantly 

different from the theoretical value, and it loses its regular 
character. This can be accounted by the following [5]: 
● the material is plastically flows in the near of the 

peaks of the profile, the bigger is the plastic strain, 
the greater is the surface roughness; 

● the workpiece and the tool are in mutual vibration 
during the cutting; 

● the flank surface of the tool is rubbing against the 
machined surface; 

● the tool edge is differing from the geometrical line 
which was taken as basis, and it has its own 
irregularities, and the tool wear is just amplifies this 
effect; 

● in case of built-up-edge the quality of the surface is 
usually deteriorating. 
 
The real (expected) value of the surface roughness can 

be calculated (forecasted) from the theoretical value [6]. 
The method for this forecasting is the following: 
● Perform experiments with the tool geometries and 

technological data according to our needs, and then 
measure the generated profile by surface tester 
equipment. 

● The connection between the measured and the 
calculated roughness data can be calculated by 
regression analysis. 
 
The most accurate approximation can be made by 

power functions in the form [7]: 

 
B

real theoretical
R A R= ⋅  (6) 

where: 
Rreal – the dependent variable; 
Rtheoretical – the independent variable; 
A, B – coefficients. 
 
It can be seen from the introduced method, that this 

is a very computing-demanded task, and therefore it is 
advisable to make a computer program for the calculation 
of theoretical values. Thus the calculation can be 
automated. The software is still under development, and 
at the time of writing this article, it is able to calculate 
the theoretical values in turning cylindrical surfaces. It 
is important to verify the theoretical calculations by 
experiments; therefore the accuracy of the prediction 
can be determined. The experimental validation of the 
introduced model will be the next step of this research 
work. 

Conslusions 

The theoretical value of surface roughness in single point 
cutting can be determined by the method introduced in 
the article for turning cylindrical surfaces, and for 
turning conical surfaces with little modifications. The 
experimental validation can confirm the accuracy of the 
model. The method can be also improved for multi-point 
cutting procedures. The calculated theoretical roughness 
values can be used to predict the real roughness values 
for a given procedure, and for the tool/material pairs 
which were experimentally investigated. 
 



 

 

193

ACKNOWLEDGEMENTS 

The described work was carried out as part of the 
TÁMOP-4.2.1.B-10/2/KONV-2010-0001 project in the 
framework of the New Hungarian Development Plan. 
The realization of this project is supported by the 
European Union, co-financed by the European Social 
Fund. The work was presented by the support of the 
Hungarian Scientific Research Fund (Number of 
Agreement: OTKA K 78482 and OTKA 84177), which 
the authors greatly appreciate. 

REFERENCES 

1. P. G. BENARDOS, G.-C. VOSNIAKOS: Predicting 
surface roughness in machining: a review, 
International Journal of Machine Tools and 
Manufacture 43, (2003), 833–844 

2. C. LU: Study on prediction of surface quality in 
machining process, Journal of materials processing 
technology 205, (2008), 439–450 

3. J. KUNDRAK: Increasing the Effectiveness of 
Machining by Application of Composite Tools in 
boring of Cylindrical and Polygon Surfaces (in 
Russian). CSc Dissertation, Tula 1986, p. 315 

4. M. C. SHAW: Metal cutting principles, Oxford 
University Press, New York, USA, 1984, p. 256 

5. J. BALI: Forgácsolás, Budapest, Tankönyvkiadó, 
1988, p. 538 

6. V. BANA: Manufacturing of high precision bores, 
PhD Dissertation, Delft, 2006, p. 149 

7. J. KUNDRAK, C. FELHO: Roughness Designeability 
of Surfaces Machined by Cutting, ICPM2009, 
September 15-19, 2009, Stara Lesna, Slovakia, 
107–112