# Mathematical Modelling of Surface Roughness Through Machining Parameters and Machining Time During the Dry Milling Process.

1. Introduction

Machinability is a fundamental technological feature of the machined metal and is an economic feature of the cutting operations. Machinability of metals and the tool cutting capacities are mutually related terms and are defined by the same method [11]

Metal cutting is one of the most significant manufacturing processes in the area of material removal. Black [2] defined metal cutting as the removal of metal chips from a workpiece in order to obtain a finished product with desired attributes of size, shape, and surface roughness [1]. One important parameter in the qualification of cut surfaces is their roughness, and its indexes. The roughness has great significance primarily at mating, sliding surfaces. This has been one more reason for the researchers' increased interest for a long time to predict these indexes for a given process within the specified cutting conditions. Several modeling procedures and techniques were worked-out, which essentially can be classified into four groups: l) analytical models, 2) experimental methods, 3) DoE (Design of Experiment)-based methods and 4) AI (Artificial Intelligence)-based methods [2, 3].

In order to establish an adequate functional relationship between the responses (such as surface roughness, cutting force, tool life/wear) and the cutting parameters (cutting speed, feed, and depth of cut), a large number of tests are needed, requiring a separate set of tests for each and every combination of cutting tool and work piece material. This increases the total number of tests and as a result the experimentation cost also increases. As a group of mathematical and statistical techniques, response surface methodology (RSM) is useful for modeling the relationship between the input parameters (cutting conditions) and the output variables. RSM saves cost and time by reducing number of experiments required [4].

Surface roughness has received serious attentions for many years. It has formulated an important design feature in many situations such as parts subject to fatigue loads, precision fits, fastener holes and esthetic requirements. In additions to tolerances, surface roughness imposes the most critical constraints for selection of machines and cutting parameters in process planning [5].

The surface finish in milling is found to be influenced in varying amounts by a number of factors, such as cutting speed, feed rate, depth of cut, material characteristics, tool geometry, workpiece deflection, stability and stiffness of the machine tool--cutting tool--workpiece system, built-up edge, cutting fluid, etc. [6].

There are various parameters used to evaluate surface roughness. In the present research for surface finish characterization in turning operations, the average surface roughness (Ra) is selected. It is the most widely used surface finish parameter in industry. Many authors suggested linear and exponential empirical models for surface roughness as functions of machining parameters by the following.

The progress in the development of predictive models, based on cutting theory, has not yet met the objective; the most essential cutting performance measures, such as, tool life, cutting force, roughness of the machined surface, energy consumption, ... etc., should be defined using experimental studies. Therefore, further improvement and optimization for the technological and economic performance of machining operations depend on a wellbased experimental methodology. unfortunately, there is a lack of information dealing with test methodology and data evaluation in metal cutting experiments [7].

Various methodologies and practices are being employed for the prediction of surface roughness, such as machining theory, classical experimental design, the Taguchi method and artificial intelligence or soft computing techniques [6].

The aim of this research is to develop the model for predict the in-process surface roughness in ball-end milling process which can be used in practice. The in-process surface roughness models are developed under various cutting conditions by employing the exponential function with the aid of the multiple regression analysis and the use of the least square method [17].

2. Nomenclature

[R.sub.a] the average surface roughness

P power

n RPM

f feed rate

[d.sub.max] workpiece diameter

L tail stock

N factorial design

K number of factors

v cutting speed

T cutting time

[c.sub.0], [c.sub.1], [c.sub.2], [c.sub.3] constants

y logarithmic value of the measured surface roughness

[[beta].sub.0], [[beta].sub.1], [[beta].sub.2], [[beta].sub.3] regression coefficients

[x.sub.0] unit vector

[x.sub.1], [x.sub.2], [x.sub.3] logarithmic values of cutting speed, feed rate, cut of depth

3. Experimental conditions

Machine tool: Universal milling machine GKA-3 motor power:8 kW, spindle speed range: 40-2000 rpm, feed rate range: 0,02-2,0 mm/rev, working are: X/Y/Z 400x2000x300 mm, max. tool diameter: 80 mm, max. tool length: 200 mm.

Workpiece was made of cold rolled steel C62D. Its chemical composition is as follows: (0.62-0.65)% C; (0.56-0.78)% Mn; 0.22% Si; 0.032% P, 0.03% S, and 98,28% Fe. Tensile strength is 230-247 N/mm2, and hardness 236-245 N/mm2. The workpiece dimensions are: the length 300 mm, the diameter 70 mm, and it is machined under dry turning conditions. Cutting tool: HSS-E Co10, with data in table 1. To guarantee the initial conditions of each test, a new tool is used in each experiment.

Roughness measuring equipment: HADRON, SRT-6210. Spectrometer Metorex Arcmet 930, Hardness meter Krautkramermic.10.DL.

4. Experimental setup

Experimental design. This work is an experimental study focused on the effect of cutting parameters and machining time on surface roughness, developing a correlation between them. The experimental design involves variation of three factors at three levels (low, medium and high), including cutting speed (v), feed rate (f) and machining time (T) as indicated in Table 2.

A design matrix was constructed on the basis of the selected factors and factor levels as shown on the table 2. The selected design matrix was a full factorial design N=2k+N0 (k=3--number of factors, N0=4--number of additional tests for three factors) consisting of 12 rows of coded/natural factors, corresponding to the number of trials. This design provides a uniform distribution of experimental points within the selected experimental hyper-space and the experiment with high resolution [7, 8].

The factor ranges were chosen with different criteria for each factor, aiming at the widest possible range of values, in order to have a better utilization of the proposed models. At the same time, the possibility of the mechanical system and manufacturer's recommendations are taken into account.

Machining conditions used in the experiment also are shown in Table 2 [9, 10, 11]. All of the trials have been conducted on the same machine tool, with the same tool type and the same cutting conditions.

Measured values of surface roughness, as the results of testing twelve experimental points defined by experiment plan matrix, are shown in Table 3. The mentioned values of surface roughness are input data for mathematical modeling of results, which was made by multiple regression analysis.

a. Regression based modeling

The main task for regression analysis is to show relationship between the roughness and machining independent variables. Many authors suggested linear and exponential empirical models for surface roughness as functions of machining parameters [13, 14, 15, 16], by the following:

[mathematical expression not reproducible] (1)

Three parameters: cutting speed (v), feed rate (f), and cutting time (T), were selected for this study, which are based on experimental results of tool life in earlier stage for the same cutting conditions [12]. Ra is the surface roughness in [micro]m, f --feed rate in mm/rev, r-nose radius in mm, T-cutting time in sec., and respectively [c.sub.0], [c.sub.1], [c.sub.2], and [c.sub.3] are constants.

Multiple linear regression models for surface roughness can be obtained by applying a logarithmic transformation that converts non-linear form of eq. (i) into following linear mathematical form:

ln [R.sub.a]= ln [c.sub.0] + [c.sub.1] ln f + [c.sub.2] ln r + [c.sub.3] ln T (2)

The linear model of eq. (3) in term of the estimated response can be written as:

y= [[beta].sub.0]+ [[beta].sub.1][x.sub.1] + [[beta].sub.2][x.sub.2]+ [[beta].sub.3]x+[epsilon] (3)

where y is the logarithmic value of the measured surface roughness, [[beta].sub.0], [[beta].sub.1], [[beta].sub.2], [[beta].sub.3] are regression coefficients to be estimated, [x.sub.0] is the unit vector, [x.sub.1], [x.sub.2], 43 are the logarithmic values of cutting speed, feed rate, cut of depth and [epsilon] is the random error.

The above equation in matrix form becomes:

y=X[beta]+ [epsilon] (4)

Thus, the least squares estimator of [beta] is

[beta] = [(X'X).sup.-1]X'y (5)

The fitted regression model is

[??]=X[beta] (6)

The difference between the experimentally measured and the fitted values of response is:

e=y- [??] (7)

The regression analysis technique using least squares estimation was applied to compute the coefficients of exponential model. The following empirical exponential model for surface roughness was determined and is given, respectively:

[R.sub.a] =1.942 [v.sup.-05426][f.sup.0.250][T.sup.0.215] (8)

5. Results and discussion

Table 3 presents experimental results of surface roughness criteria [R.sub.a] for various combinations of cutting speed, feed rate and machining time to full factorial design. Minimal value of surface roughness criteria [R.sub.a] was obtained at V = 94.20 m/min, f = 0,1 mm/rev, T=900 s, (test No. 8). That means increasing of cutting speed with the lowest feed rate and machining time lead to decreasing of surface roughness.

It is found that cutting speed has the most significant effect on surface roughness, followed by feed rate and machining time.

Maximal value of surface roughness criteria [R.sub.a] was registered at V = 56.52 m/min, f = 0.3 mm/rev, T =1200 s, (test No. 2). In order to achieve better surface finish, the highest level of cutting speed, and the lowest level of feed rate and machining time, should be recommended.

The analysis of obtained mathematical models and cutting data influence at surface roughness has been made using diagrams shown in Figures 4, 5, 6.

Fig. 1 which highlights the main factor plots for Ra appears to be an almost linear decreasing function of cutting speed and an increasing function of feed rate (f) and cutting time (T).

Figs. 2 a, b, c and d illustrate 3D surface plots of Ra according to the predictive exponential empirical model (9).

6. Conclusions

This paper presents research of various cutting parameters affecting the surface roughness in dry milling of carbon steel using HSS cutters. The investigations of this study indicate that the cutting parameters like cutting speed and feed rate are the primary influencing factors, which affect surface roughness.

Statistical models deduction defined the degree of influence of each cutting regime element on surface roughness criteria. The results revealed that cutting speed seems to influence surface roughness (0.542) more significantly than feed rate (0.250). However, machining time is less significant (0.215). With the regression equation generated, the best combination of design independent variables for achieving the optimization of cutting processes.

The relations of the surface roughness, the cutting speed, the feed rate, and the cutting time are investigated to develop the prediction models of surface roughness. The exponential function is employed to represent the relation of the arithmetic average surface roughness, the cutting force ratio, and the cutting parameters. The multiple regression analysis has been utilized to calculate the regression coefficients of the in-process prediction of surface roughness model by using the least square method.

DOI: 10.2507/27th.daaam.proceedings.028

7. References

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[7] N. Qehaja, H. Zeqiri, A. Salihu, H. Osmani, F. Zeqiri, Machinability of Matals, Methods, and Practical Application, Annals of DAAAM & Proceedings 2012, The 23rD World Symposium, 24-27th October 2012, University of Zadar, Croatia, URL: http:Wwww.daaam.com, Vienna University of Technology, Karlsplatz 13/311, A-1040 Wien, Austria, Europe

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[10] N. Qehaja, A. Bunjaku, J. Kacani. A.Salihu, H. Zeqiri, H. Osmani, Mathematical models of flank wear curve during the turning of steel C60E4, The 20th International DAAAM SYMPOSIUM "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions" 22- 25th October Vienna, Austria, 2009

[11] N. Qehaja, A. Salihu, H. Zeqiri, H. Osmani & F. Zeqiri, Machinability of metals, methods and practical application, Annals of DAAAM for 2012 & Proceedings of the 23rd International DAAAM Symposium, Volume 23, No. 1, ISSN 2304-1382 ISBN 978-3-901509-91-9, CDROM version, Ed. B. Katalinic, Published by DAAAM International, Vienna, Austria, EU, 2012

[12] H. Zeqiri, N. Qehaja, F. Zeqiri, A. Salihu, H. Osmani, Research of Flank Wear in Turning of Steel Ck 60, Annals of DAAAM & Proceedings 2012, The 23rd DAAAM World Symposium, 20012

[13] A. Salihu, H. Zeqiri, A. Bunjaku, N. Qehaja, H. Osmani, A. Kygyky, Research of the horizontal parameters of the roughness machined surface by turning, The 20th International DAAAM Symposium "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solution", 2009

[14] M. Durairaja, S. Gowri, Parametric Optimization for Improved Tool Life and Surface Finish in Micro Turning using Genetic Algorithm, International Conference On DESIGN AND MANUFACTURING, IConDM 2013, Procedia Engineering 64 (2013) 878-887

[15] S. Chinchanikar, A.V. Salve, P. Netake, A. More, S. Kendre and R. Kumar, Comparative evalutions of surface roughness during hard turning under dry and with water- based and vegetable- oil-based cutting fluids, International Conference on Advances in Manufacturing and Materials Enginneering AMNE 2014, Procedia Materials Science 5 (2014) 1966-1975

[16] N. Qehaja, K. Jakupi, A. Bunjaku, M. Brugi, H. Osmani, Effect of machining parameters and machining time on surface roughness in dry turning process, 25th DAAAM International Symposium on Intelligent Manufacturing and Automation, DAAAM 2014, Procedia Eng 100:135-140, 2015

[17] S. Tangjitsitcharoen, A. Senjuntichai, In- Process monitoring and prediction of surface roughness in ball-end milling process, Annals of DAAAM for 2010 & Proceedings of the 21st International DAAAM Symposium, Volume 21, No. 1, ISSN 1726-9679 ISBN 978-3-901509-73-5, Editor B. Katalinic, Published by DAAAM International, Vienna, Austria, EU, 2010

This Publication has to be referred as: Qehaja, N[exhat]; Doci, I[lir]; Bruqi, M[irlind]; Abdullahu, F[itore]; Jakupi, K[altrina] & Zhujani, F[atlume] (2016). Mathematical Modelling of Surface Roughness Through Machining Parameters and Machining Time During the Dry Milling Process, Proceedings of the 27th DAAAM International Symposium, pp.0187-0194, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3- 902734-08-2, ISSN 1726-9679, Vienna, Austria

Caption: Fig. 1. The dependence of surface roughness on: a) cutting speed and various values of cutting time, b) cutting speed and various values of feed rate, c) feed rate and various values of cutting speed, d) feed rate and various values of cutting time

Caption: Fig. 2. 3D surface plots: Plot3D [6.795*(56.52[conjunction]0.542)*(f[conjunction]0.250)*(T[conjunction]0.215),{f, 0.1,0.3},{T,900,1200}]
```Table 1. Cutting tool data

DIN 844        [d.sub.1]    [d.sub.2]    [1.sub.1]    [1.sub.2]    z
[mm]         [mm]         [mm]         [mm]

HSS-E Co10         20           20          104           38       3

Table 2. Experimental setup at three level factors.

Cutting factors and their
levels

High    Middle    Low
level   level    level
No.    Factors        Code
level        1       0       -1

1    v, m/min     [X.sub.1]    94.20   72.96    56.52
2    f, mm/rev    [X.sub.2]     0.3     0.18     0.1
3    T, s         [X.sub.3]    1200     1049     900

Table 3. Experimental results.

Coded factors

Test    [X.sub.0]    [X.sub.1]    [X.sub.2]    [X.sub.3]
No.

1          +1           +1           -1           +1
2          +1           -1           +1           +1
3          +1           +1           +1           -1
4          +1           -1           -1           -1
5          +1           0            0            0
6          +1           0            0            0
7          +1           -1           -1           +1
8          +1           +1           -1           -1
9          +1           +1           +1           +1
10         +1           -1           +1           -1
11         +1           0            0            0
12         +1           0            0            0

Performance measures

Test      Ra      Y = ln R a
No.

1      2.783    1.023529483
2      2.985    1.093599747
3      1.982    0.684106436
4      1.524    0.421338457
5      1.963    0.674473915
6      1.882    0.632335041
7      1.479    0.391366184
8      1.256    0.227932068
9      1.145    0.135404637
10     3.182     1.15750993
11     1.714     0.53882982
12     1.825    0.601579987
```