Advances in Production Engineering & Management

ISSN 1854-6250

Volume 16 | Number 3 | September 2021 | pp 326–334

Journal home: apem-journal.org

https://doi.org/10.14743/apem2021.3.403

Original scientific paper

Modeling and optimization of finish diamond turning of
spherical surfaces based on response surface methodology
and cuckoo search algorithm
Kramar, D.a,*, Cica, Dj.b
aUniversity

bUniversity

in Ljubljana, Faculty of Mechanical Engineering Ljubljana, Slovenia
in Banja Luka, Faculty of Mechanical Engineering Banja Luka, Bosnia and Herzegovina

ABSTRACT

ARTICLE INFO

Surface roughness is one of the most significant factors to indicate the product quality. Diamond turning is an efficient and highly accurate material removal process to improve the surface quality of the workpiece. In the present
study, the arithmetic mean absolute roughness (Ra) and total height of profile
(Rt) of spherical surface during finish turning of a commercial brass alloy
CuZn40Pb2 were modeled using Response Surface Methodology (RSM). The
experimental investigations were carried out using the Central Composite
Design (CCD) under dry conditions. The effect of cutting parameters such as
spindle speed, feed rate and depth of cut) on spherical surface quality was
analyzed using analysis of variance (ANOVA). A cuckoo search (CS) algorithm
was used to determine the optimum machining parameters to minimize the
surface roughness. Finally, confirmation experiments were carried out to
verify the adequacy of the considered optimization algorithm.

Keywords:
Brass alloy;
Diamond turning;
Surface roughness;
Spherical surface;
Modeling;
Optimization;
Response surface methodology
(RSM);
Cuckoo search (CS)
*Corresponding author:
davorin.kramar@fs.uni-lj.si
(Kramar, D.)

Article history:
Received 31 May 2021
Revised 3 September 2021
Accepted 15 September 2021
Content from this work may be used under the terms of
the Creative Commons Attribution 4.0 International
Licence (CC BY 4.0). Any further distribution of this work
must maintain attribution to the author(s) and the title of
the work, journal citation and DOI.

1. Introduction
Brass (copper-zinc) alloys are used in a wide range of industrial applications such as mechanical,
electrical and hydraulic systems due to their excellent formability, high thermal and electrical
conductivity, corrosion resistance and excellent antibacterial properties. To improve machinability, various alloying elements are commonly added to brass. The most important of these elements is lead, which improves machinability in terms of tool wear, cutting forces, chip breaking
and surface quality [1-3]. Extensive research has been carried out in recent decades to improve
the machinability of brass alloys.
In [4], a comparison of the machinability of three lead-free brass alloys and one leaded brass
alloy in terms of energy consumption and chip morphology was carried out. The influence of
different coating types and the use of diamond tools on cutting forces, chip formation and sur326

Modeling and optimization of finish diamond turning of spherical surfaces based on response surface methodology and …

face quality was investigated by Klocke et al. [5]. Nobel et al. [6] analyzed the chip formation
process in different low lead brass alloys. Schultheiss et al. [7] analyzed cutting forces, surface
quality and tool wear during longitudinal turning to evaluate the machinability of leaded and
lead-free brass alloys. The effects of minimum quantity lubrication (MQL) and cutting parameters on surface quality during turning of commercial brass were studied by Davim et al. [8]. A
comparison with conventional flooding conditions was also made. The machinability of the highly leaded brass alloy CuZn39Pb3 [9] and commercial lead-free brass alloys [10] was also analyzed. Vilarinho et al. [11] studied the influence of the chemical composition of brass alloys on
surface quality and machining forces during turning. Schultheiss et al. [12] compared the machinability and manufacturing costs in turning of conventional leaded brass alloys and a lowlead alternative. Toulfatzis et al. [13] studied the chip morphology and tool wear during longitudinal turning of two leaded brass alloys. The Taguchi method with the utility concept was introduced for simultaneous optimization of surface quality and specific cutting force in turning of
brass CuZn39Pb3 under MQL cutting [14]. Toulfatzis et al. [15] used the signal-to-noise ratio for
single optimization of surface roughness and cutting force in longitudinal turning of lead-free
and leaded brass alloys.
Some researchers also used artificial intelligence based methods. Gaitonde et al. [16] applied
a genetic algorithm to determine the optimum machining parameters for minimizing surface
roughness when turning leaded brass alloys under MQL conditions. Raja and Baskar [17] presented a multi-objective optimization method based on particle swarm optimization with two
objectives, namely machining time and surface quality. Optimum cutting conditions during turning for different materials namely brass, aluminum, mild steel and copper were selected. Natarajan et al. [18] estimated the surface roughness in longitudinal turning of brass using artificial
neural networks.
After a review of the literature, it is apparent that a considerable amount of research has been
conducted on turning of brass alloys. However, none of the literature reviewed dealt with finish
diamond turning of spherical surfaces. As market competitiveness increases, surface roughness
is probably one of the most widely used indicators of surface quality of machined parts today
[19-27]. Surface roughness affects several functional properties of products, especially friction,
fatigue strength, wear, heat transfer, light reflection, lubricant distribution, etc. Ensuring surface
quality is one of the most critical issues in the fully automated mass production of parts with
spherical surfaces. However, surface roughness is strongly influenced by the variation of process
parameters and analytical modeling is difficult due to its nonlinearity. Therefore, it is imperative
to develop a mathematical model of surface roughness, exploit the influence of machining parameters, and finally optimize the surface quality in brass ball turning.

2. Experimental procedure

The machining tests were performed on the special machine tool Picchi Diamantatrice to produce spherical parts with a spindle power of 5.5 kW and a maximum spindle speed of 7000 rpm.
This lathe was equipped with two tools: a carbide turning tool for roughing and a diamond tool
for finishing and machines spheres with a diameter of 2" (Fig. 1). Commercially available brass
CuZn40Pb2 (CW617N) workpieces were used for machining. The chemical composition of the
material is summarized in Table 1. The working material has a hardness of 90-100 HRB and a
tensile strength of 390-440 N/mm2. This alloy provides good corrosion resistance together with
good formability.
Alloy

CuZn40Pb2

Cu (%)
57-59

Table 1 Chemical composition of the studied brass alloys

Zn (%)
rest

Pb (%)
1.6-2.5

Al (%)

0.05 max

Fe (%)

0.3 max

Advances in Production Engineering & Management 16(3) 2021

Ni (%)

0.3 max

Sn (%)

0.3 max

Others: Mn, Si, Sb, As, Bi,
(%)
0.2 max

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Kramar, Cica

The diamond cutting tool was used for precision machining of a ball valve. The tool nose does
not have a radius like conventional cutting tools, but a flat nose that allows very fine machining.
The dimensions of the tool were 95 × 12 × 12 mm, with the sides at a 45° angle and a 1.3 mm
wide cutting edge. The main angles of the tool cutting wedge were 2° for clearance angle α, 10.5° for rake angle γ and consequently 98.5° for wedge angle β. The cutting depth of the diamond segment is 0.4 mm and represents the maximum theoretical cutting depth, but due to the
possibility of pulling the diamond segment out of the tool holder, the depth of cut is usually not
used deeper than 0.2 mm.

Fig. 1 Experimental setup for diamond fine turning of CuZn40Pb2

The arithmetic mean of the absolute roughness (Ra) and a total height of profile (Rt) of the machined workpiece were measured with the surface roughness measuring device Tesa Rugosurf
20 (Fig. 2). The examined length was 3.2 mm with a basic span of 0.8 mm.

Fig. 2 Surface roughness measurements

In the present study, three cutting parameters, i.e., spindle speed (n), feed rate (f) and depth
of cut (a) were selected, and the ranges of machining conditions were defined through initial
experiments. The cutting parameters and their levels are given in Table 2.
The experiments were designed and carried out according to the Central Composite Design
(CCD). All experiments were conducted under dry conditions. The measured values of Ra and Rt
are shown in Table 3.
Parameter

Spindle speed, n
Feed, f
Depth of cut, a

328

Table 2 Machining parameters and their levels
Levels
Unit
Level 1
Level 2
min-1
4800
5500
mm/rev
0.48
0.64
mm
0.075
0.125

Level 3
6200
0.80
0.175

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Modeling and optimization of finish diamond turning of spherical surfaces based on response surface methodology and …

No.

n (min-1)
5500
4800
6200
5500
4800
4800
5500
6200
5500
4800
5500
5500
4800
6200
6200
6200
5500

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

Table 3 Experimental layout for central composite design
Process parameters
Response measurements
f (mm/rev)
a (mm)
Ra (µm)
Rt (µm)
0.64
0.125
0.140
0.97
0.48
0.075
0.208
1.38
0.80
0.075
0.164
1.11
0.48
0.125
0.173
1.18
0.48
0.175
0.212
1.76
0.80
0.075
0.249
2.10
0.64
0.125
0.154
1.00
0.64
0.125
0.155
1.07
0.80
0.125
0.152
1.11
0.64
0.125
0.179
1.29
0.64
0.125
0.156
1.06
0.64
0.075
0.149
1.05
0.80
0.175
0.224
1.56
0.80
0.175
0.150
1.14
0.48
0.175
0.215
1.56
0.48
0.075
0.145
1.35
0.64
0.175
0.218
1.51

3. Results and discussion

The reduced quadratic polynomial model, given as a function of the machining parameters for an
arithmetic mean of the absolute roughness (Ra) and a total height of the profile (Rt) using the
Response Surface Methodology (RSM) is represented by equations (1) and (2), respectively
𝑅𝑅𝑎𝑎 = 0.4063 − 3.4714 ∙ 10−5 𝑛𝑛 + 0.212𝑓𝑓 − 2.1591𝑎𝑎 − 1.7656𝑓𝑓𝑓𝑓 + 13.9886𝑎𝑎2

𝑅𝑅𝑡𝑡 = 8.5175 − 3.2011 ∙ 10−3 𝑛𝑛 + 9.2605𝑓𝑓 − 14.7502𝑎𝑎 − 1.317 ∙ 10−3 𝑛𝑛𝑛𝑛 − 17.1875𝑓𝑓𝑓𝑓
+ 3.4347 ∙ 10−7 𝑛𝑛2 + 107.3208𝑎𝑎2

(1)
(2)

The adequacy of the developed models was examined using analysis of variance (ANOVA).
The results are given in Table 4 and Table 5 for Ra and Rt, respectively. The F-values for Ra and Rt
are 5.80 and 7.85, respectively. For the desired confidence level (95 %), the F-values of the established model exceed the F-value of the standard tabulated F-values for Ra and Rt of 3.204 and
3.293, respectively. Thus, the two reduced quadratic models can be considered appropriate
within the confidence limit. P-values smaller than 0.05 imply that n and the product a2 are statistically significant terms for the arithmetic mean of absolute roughness Ra, while n and the products n × f, f × a and a2 are also observed to be significant terms for the total height of the profile
Rt. Other parameters have no statistical significance for the surface quality responses considered. The squared correlation coefficient (R2) values of 0.7250 and 0.8593 for Ra and Rt, respectively, show good agreement among the experimental and predicted values for both models.
The 3D response surfaces were also created to study the effects of the machining parameters
and their interactions. Figs. 3-5 show the contour plots for the surface roughness parameters (Ra
and Rt) as a function of spindle speed (n), feed rate (f) and depth of cut (a).
Source
Model
n
f
a
f ×a
a2
Error
Total

Sum of squares
0.014
5.905·10-3
1.96·10-5
1.082·10-3
1.596·10-3
5.036·10-3
5.172·10-3
0.019

Table 4 The ANOVA table for Ra
DF
Mean square
5
2.728·10-3
1
5.905·10-3
1
1.96·10-5
1
1.082·10-3
1
1.596·10-3
1
5.036·10-3
11
4.072·10-4
16

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F-value
5.80
12.56
0.042
2.30
3.39
10.71

P-value
0.0073
0.0046
0.8419
0.1575
0.0925
0.0074

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Kramar, Cica

Source
Model
n
f
a
n×f
f×a
n2
a2
Error
Total

Sum of squares
1.31
0.35
4.41·10-3
0.029
0.17
0.15
0.086
0.22
0.21
1.52

Table 5 The ANOVA table for Rt
DF
Mean square
7
0.19
1
0.35
1
4.41·10-3
1
0.029
1
0.17
1
0.15
1
0.086
1
0.22
9
0.024
16

F-value
7.85
14.53
0.19
1.22
7.31
6.35
3.60
9.16

P-value
0.0031
0.0041
0.677
0.2971
0.0242
0.0327
0.0902
0.0143

The effects of spindle speed and feed rate on roughness parameters are shown in Fig. 3,
where the depth of cut is kept at an intermediate level. The minimum value of Ra was obtained at
high spindle speed and low feed rate, as shown in Fig. 3a. It should also be noted that the Ra value is almost directly proportional to these two parameters, with feed rate having less significance. Fig. 3b shows that Rt is significantly affected by spindle speed, while feed rate has less
influence on it. The minimum value of Rt was also found at the maximum values of spindle speed
and feed rate. Therefore, to obtain better roughness parameters, higher spindle speeds and feed
rate should be preferred for diamond finish turning of spherical surfaces.

Fig. 3 Surface plot of Ra (a) and Rt (b) with spindle speed and feed rate

Fig. 4 shows the estimated response surface with respect to the spindle speed and the depth
of cut, while the feed rate is kept at an intermediate level. The minimum values of both parameters were obtained at high spindle speed and medium depth of cut. It can also be observed that
both cutting parameters strongly influence the surface quality.

Fig. 4 Surface plot of Ra (a) and Rt (b) with spindle speed and depth of cut

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Modeling and optimization of finish diamond turning of spherical surfaces based on response surface methodology and …

Fig. 5 shows the influences of feed rate and depth of cut on the surface roughness parameters
(Ra and Rt), while the spindle speed is kept at an intermediate level. The plots show that the minimum values for both parameters were found at a high feed rate, while the depth of cut was
about 0.1 mm. This figure shows that both roughness parameters are almost directly proportional to the feed rate. It is also worth noting that the depth of cut is more significant compared
to the feed rate.

4. Optimization

Fig. 5 Surface plot of Ra (a) and Rt (b) with feed rate and depth of cut

4.1 Cuckoo search
The cuckoo search (CS) algorithm, originally introduced by Yang and Deb [28], is one of the recent swarm intelligence-based optimization algorithms. The CS algorithm was inspired by the
brood parasitism behavior of certain cuckoo bird species and the characteristics of Lévy flights
discovered in the flight behavior of numerous insects and animals. Female cuckoos lay their eggs
in the nests of other, typically different species of host birds. These eggs also resemble the host
birds' eggs in color and pattern. When the host bird realizes that the eggs are not its own, it either discards them or simply leaves the nest and builds a new one elsewhere. Cuckoos must
therefore mimic their host birds' eggs very closely to minimize the likelihood that their eggs will
be abandoned. Optimization is about replacing a less good nesting solution with a new and potentially better one (cuckoo).
The CS algorithm is essentially based on three idealized rules [29]: (i) each cuckoo lays one
egg at a time in a randomly selected nest; (ii) an elite selection procedure is used in which only
the best nests with superior quality eggs are passed on to the next generation; (iii) the number
of available host nests is fixed and a host bird can detect a foreign egg with probability in the
range of [0, 1].
While producing new solutions x(t+1) for the i-th cuckoo, Lévy flight is performed
(𝑡𝑡+1)

(𝑡𝑡)

+ 𝛼𝛼 ⊗ Lévy(𝜆𝜆)

(3)

Lévy ∼ 𝑢𝑢 = 𝑡𝑡 −𝜆𝜆 , (1 < 𝜆𝜆 ≤ 3)

(4)

𝑥𝑥𝑖𝑖

= 𝑥𝑥𝑖𝑖

where x(t)i denotes the i-th candidate solution at iteration t and α > 0 denotes the step size factor.
In most cases, α = 1 can be used. The product ⊗ denotes inputwise walking for multiplications.
Basically, the Lévy flight results in a random walk, where the random step length is determined by a Lévy distribution with infinite mean and infinite variance
4.2 Optimization model

The optimization process in this paper aims to find the best combination of process parameter
levels that results in the lowest value of Ra and Rt. To formulate the optimization problem, the
regression models for Ra (Eq. 1) and Rt (Eq. 2) were used as the objective function. In this study,
Advances in Production Engineering & Management 16(3) 2021

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Kramar, Cica

three variables, namely spindle speed, feed rate and depth of cut were considered as optimization variables. Normalization of each subobjective was also introduced to compensate the differences in numerical values between them. Thus, the resulting objective function (ROF) to be minimized is a weighted combination of the two objectives as follows:
ROF (𝑛𝑛, 𝑓𝑓, 𝑎𝑎) = 𝑤𝑤1

𝑅𝑅𝑎𝑎

𝑅𝑅𝑎𝑎min

+ 𝑤𝑤2

𝑅𝑅𝑡𝑡

(5)

𝑅𝑅𝑡𝑡min

where w1 and w2 are the weight factors of Ra and Rt, respectively. In this paper, equal weights for
Ra and Rt were selected, i.e. w1 = w2 = 1/2.
The minimization of the ROF (Eq. 5) is subject to the limits of the cutting parameters. The
boundary conditions were the upper and lower limit of the experimental machining parameters
(Table 2) and were given as follows: 4800 ≤ n ≤ 6200 min-1, 0.48 ≤ f ≤ 0.8 mm/rev, 0.075 ≤ a ≤
0.175 mm.
After formulating the optimization problem and its constraints, the CS optimization algorithm
was employed to solve the problem. The proposed CS algorithm requires some setting parameters for implementation. The minimum value of the resulting objective function (ROF = 0.916)
was obtained for a population size of 20, termination probability of 0.25, and termination criterion of 100 generations. The results of the CS optimization algorithm showed that the best combination of turning parameters for simultaneous optimization of the arithmetic mean of absolute
roughness (Ra) and total height of profile (Rt) was: 6200 min-1 for spindle speed, 0.8 mm/rev for
feed rate, and 0.13 for depth of cut.
4.3 Confirmation experiments

The confirmation tests were carried out at the optimum values of the process parameters to
verify the quality characteristics of the spherical fine turning process recommended in the study.
In accordance with the obtained optimum results, four new experiments were carried out (Table
6). The mean values of Ra and Rt were 0.139 µm and 1.04 µm, respectively, which were in good
agreement with the predicted values. Consequently, the proposed CS optimization algorithm
was efficient to find out the optimal set of machining parameters for spherical finish turning
associated with minimum surface roughness.
Confirmation test
Ra (µm)
Rt (µm)

5. Conclusions

1
0.143
1.19

Table 6 Results of confirmation tests
2
3
0.149
0.134
0.89
1.10

4
0.141
0.97

Mean
0.139
1.04

In the present research work, the influence of machining parameters such as spindle speed, feed
rate and depth of cut on the arithmetic mean of absolute roughness (Ra) and total height of profile (Rt) in diamond finish turning of brass CuZn40Pb2 was investigated. The following conclusions can be drawn from the analysis of the results, subsequent model development and optimization:
• The response surface methodology in combination with the central composite design has
been successfully applied to study the effects of different machining parameters on two
quality characteristics, namely the arithmetic mean of the absolute surface roughness
and the total profile height, during diamond finish turning of spherical surfaces. The developed mathematical models of both responses in terms of the actual design factors,
their interactions and quadratic terms are suitable for the analysis of the sphere turning
process.
• The spindle speed and the quadratic term of the depth of cut were the significant parameters affecting the arithmetic mean of the absolute roughness, according to the results of
ANOVA. The ANOVA also showed that the spindle speed, the quadratic term of the depth
of cut, the interaction terms between the spindle speed and the feed rate, and the feed
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Modeling and optimization of finish diamond turning of spherical surfaces based on response surface methodology and …

•

•

rate and the depth of cut were significant terms affecting the performance of the total
height of the profile. Spindle speed showed the greatest influence on both surface parameters compared to depth of cut and feed rate.
The best combination of cutting parameters to simultaneously minimize the arithmetic
mean of the absolute roughness and the total height of the profile, obtained by the optimization model based on the cuckoo search algorithm, was as follows: 6200 min-1 for
spindle speed, 0.8 mm/rev for feed rate and 0.13 mm for depth of cut. This study and the
determined combination of parameters allow a more detailed technology planning for
the production of high-quality spherical components, both from the point of view of premachining and productivity itself. The results obtained do not confirm the theory of
turning. In fact, in our case, we obtain the best results at the highest feed rate, which is
contrary to expectations. This is certainly related to the kinematics and stability of the
machine tool in this speed and feed range, which in turn shows how important it is to optimize each machining process.
The results of the confirmation experiments reveal the effectiveness of the proposed algorithm for optimizing the surface quality characteristics in diamond finish turning of
spherical surfaces.

Acknowledgement

The research was conducted within the framework of a scientific and technological intergovernmental bilateral project supported by the Slovenian Research Agency under the project serial numbers: P2-0266 and BI-BA/19-20-008,
and by the relevant ministries Government of the Republic of Srpska, Bosnia and Herzegovina.

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