1. Introduction
Nowadays, titanium and its alloys are widely used in various areas, such as the aerospace, medical and automotive industries, due to their excellent properties (e.g., high strength-to-weight ratio and good corrosion resistance, relatively low density, high-temperature properties, excellent creep, biocompatibility) [
1]. Vanadium, molybdenum, manganese and aluminum are often alloying elements and provide high strength [
2]. However, titanium alloys are classified as difficult-to-cut materials. The poor machinability of the materials is caused by their properties [
3]. A rapid tool wear rate due to the low thermal conductivity and high chemical reactivity causes high cutting temperature at the cutting zone [
4].
Difficult-to-cut materials generally make it challenging to obtain the required surface integrity, high performance and economic of machining [
5]. The indicators depend mainly on the kind of machined workpiece and tool life [
6]. Various methods and technologies have been developed to improve the quality of machined surface and to increase performance of machining [
7]. Additionally, the decrease of manufacturing costs plays an important role. Necessary optimization should simultaneously provide a short machining time and obtain the required quality of surface roughness. To minimize costs and increase performance machining, application of different values of cutting data can be used.
Nowadays, to optimize machining process parameters, various evolutionary or meta-heuristic methods can be used, such as GA, PSO, ACO and ABC. The application of the techniques in optimizing machining process parameters has been proven in the literature [
8]. Although these methods are applied in many practical cases, they characterize limitations related to their inherent search mechanism. The solutions of the techniques depend generally on the type of objective and constraint functions (linear, non-linear, etc.) and the type of variables used in the problem modeling (integer, binary, continuous, etc.). To predict the performance of machining processes, regression models based on experimental tests have been developed. These regression models can be solved by using traditional optimization methods which are sensitive to the initial assumption. Excellent solutions in the case of several input parameters is difficult [
9].
In manufacturing technology, surface roughness is an important indicator that can affect the performance of mechanical parts (product wear, fatigue strength, tribological properties, corrosion resistance [
10] and manufacturing cost [
4]. Therefore, evaluating surface roughness parameters is significant. Hence, optimization is widely used to achieve the required surface quality and process performance. One of the numerous methods for surface roughness prediction and obtaining optimal cutting parameters is the Response Surface Methodology (RSM) [
11].
In the turning process (one of the most used in manufacturing technology), the values of machining parameters (feed rate, cutting speed, and cutting depth), nose radius, cutting time, cutting fluid, and cutting forces are subjected to optimization [
2,
12]. The cutting data directly affects the surface roughness, dimensional accuracy, tool wear rate, machining performance and manufacturing costs. The selection of appropriate parameters in order to improve surface finish is difficult [
13,
14,
15].
In the case of turning of titanium alloy, many researchers analyze the impact of machining parameters on surface roughness and optimize their values to provide the required surface quality. The authors [
2] have developed mathematical models to predict the surface roughness after the turning process of titanium alloy. The impact of the cutting parameters and the kind of tool materials on the surface roughness were examined. The optimum conditions obtained for uncoated tools are cutting speed
vc = 80 m/min, feed
f = 0.05 mm/rev, and depth of cut
ap = 0.25 mm. The authors proved that optimization techniques and mathematical models reduced the cost of machining. The proposed optimum process parameters resulted in an increase of surface finish. In [
4], cutting parameters such as feed rate, cutting speed, and depth of cut were used to predict surface roughness in turning of aerospace titanium alloy (gr5). The proposed model studied the effect on surface roughness when varying the turning parameters using the surface plots. The analysis of results showed the feed rate to be the most influential parameter on the surface roughness. Yang at al. [
11] present the prediction model to predict the surface roughness and cutting parameters in the turning process of TC11 titanium alloy. It also indicates that the feed rate is the most important parameter influencing surface roughness, followed by cutting speed. The cutting depth has minimal effect.
In the literature, different approaches of prediction surface roughness in machining have been presented. Asiltürk et al. [
16] used the Taguchi experimental test to design optimized turning parameters and to obtain the lowest degree of surface roughness parameters (Ra and Rz). The results of the study showed that the most influential factors included the feed rate and the interaction between feed rate and cutting speed over the surface roughness. Makadia et al. [
17] proposed developing the surface roughness prediction model of AISI 410 steel with the aid of a statistical method under various cutting conditions, such as cutting speed, feed rate, depth of cut and tool nose radius. The results analysis showed that the feed rate is the main influencing factor on the roughness, followed by the tool nose radius and the cutting speed. Depth of cut proved to be an insignificant parameter on the surface roughness. Yamane et al. [
18] presented a method for quantitatively estimating the cutting stability and the machining system stability in the turning process. This method used the machined surface roughness profile to evaluate the position of the cutter in order to obtain the cutting edge transferability and the stability in the feed direction and in the depth-of-cut direction. Three samples with different roughness were estimated using the proposed evaluation method. The results showed that it is possible to quantitatively evaluate cutting instability based on adhesion or built-up edge, as well as the system instability resulting from vibration during machining. In [
19], the authors presented a method of predicting roughness profile by further developing a methodology linking the theoretically developed models to the real machining process conditions, which are different for different processing systems (machines). This is significant, because researchers usually focus on finding prediction models for the Ra and Rz (Rt) parameters. In this approach, the statistical equality of sampling lengths in surface roughness measurement proves to be a major parameter that provides information about the condition of the process with respect to surface roughness formation. The authors demonstrated that the indicator of the roughness profile condition can be successfully implemented in roughness profile prediction. Özel and Karpat [
20] focused on the development of models based on feedforward neural networks in accurately predicting both surface roughness and tool flank wear in finish dry hard turning. The neural network models were trained based on the experimental data of measured surface roughness and tool flank wear. The authors assumed that the neural network models provided better prediction capabilities because they were able to model more complex nonlinearities and interactions in comparison to linear and exponential regression models. In [
21], a model was proposed based on ANN (Artificial Neural Network) to predict surface roughness (Ra) in terms of cutting parameters (such as feed rate, cutting speed, depth of cut) during hard turning of AISI H13 tool steel with minimal cutting fluid application. The authors showed that the ANN model could be applied successfully in fixing the cutting parameters to achieve desired surface finish and to maintain the surface finish within the tolerance limits during automated hard turning of AISI H13 steel with minimal fluid application.
Nowadays, machined parts have more and more complicated and nonlinear shapes. However, there is a lack of prediction models of curvilinear surface roughness in the turning process. There is a need to provide aligned values of roughness parameters on the whole surface after the turning process. The task is especially difficult when difficult-to-cut materials are machined. The prediction of surface roughness parameters values is a challenge that can result from the variable impact of cutting forces and change in the surface tilted angle.
In this study, a new optimized method is presented that involves the prediction of the curvilinear surface roughness. The proposed method was formulated based on the experimental research results. The created model also results in a short machining time and low manufacturing cost. In the first part of the study, the results of experimental tests of turning six curvilinear surfaces made of Ti6Al4V alloy are presented. Selection of the cutting parameters plays an important role in achieving high cutting performance and the required surface roughness. The experimental research was focused on the impact of cutting speed and feed rate on the quality of the spherical surface (values of roughness parameters Ra and Rz) for various surface tilted angles. The second part of the paper presents the optimization procedure for obtaining aligned surface roughness parameters for the example spherical profile of the part.
3. Case Study—Method Verification
The verification of the proposed method was performed based on an experimental test. During the experiment, the surface of a Ti6-Al-4V alloy part, marked by the red line in
Figure 12, was machined.
The selected surface was machined by using the cutting insert with symbol N123F1-0318-RO S05F. In the first part of the research, the following cutting data with constant values was applied: depth of cut
ap = 0.7 mm, feed rate
f = 0.085 mm/rev and cutting speed
vc = 80 m/min. The surface was characterized by the required surface roughness class IT8 (Ra = 0.63 μm, Rz = 3.2 μm). On the machined surface, changes in the surface roughness values were observed. The changes were analogous to those described in
Section 2.1 and
Section 2.2.
The second part of the research consisted of generating and optimizing the NC code according to the algorithm presented in
Figure 11. Due to the possibility of saving only one feed rate in the one code line (one tool movement), the analyzed contour of the part was divided into smaller fragments with a length 0.5 mm. Next, the tilted angle
δI of the contour and the initial parameters (such as:
fmin = 0.03 mm/rev,
fmax = 0.14 mm/rev, Ra
max = 0.63 μm and Rz
max = 3.2 μm) were determined. In the next step, feed rate optimization was performed on the basis of the mathematical model, presented in the
Section 2.2 (Equations (3) and (4)). The calculated feed rate values (
fRa_OPTi,
fRz_OPTi and
fOPTi) were compared with the limit values
fmin and
fmax. In
Figure 13, the optimized feed rate values
fOPT and the marked contour of the machined surface are presented.
As a result of the feed rate optimization, the desired goal was achieved, meaning that the values of the surface roughness parameters were below their upper limit values (Ra ≤ 0.63 μm and Rz ≤ 3.2 μm) regardless of the surface shape and its tilted angle. The average values with standard deviations of the surface roughness parameters Ra and Rz, measured in points such as A–I (according to
Figure 13), are shown in
Figure 14.
During the experimental tests, the measurements of 3D surface roughness were carried out. Examples of the topographies and isometric views of points such as B, D, E, and G (according to
Figure 13), are shown in
Figure 15.
5. Conclusions
The research analysis presented in this paper concerns the significant problem of the locally variable roughness of curvilinear surfaces occurring after turning. This problem appears to be important from the point of view of the quality of the surface of manufactured machine parts. The authors of the paper have developed a mathematical model for predicting the values of the curvilinear surface roughness parameters Ra and Rz. The method for optimization of cutting data (cutting speed and feed rate) for machining the curvilinear surfaces was proposed by taking into account the alignment of the surface roughness parameters. The research indicated the insignificant effect of the cutting speed on the surface roughness parameters. This means that the cutting speed value has been correctly selected in the investigations, meaning that in this case, the regression equation can be simplified to optimize only the feed rate and tilted angle.
The case study presented in the second part of the paper verified the correctness of the developed machining strategy. According to the proposed optimization method, the machining time of the example curvilinear surface of the titanium alloy workpiece was shortened by almost half in comparison to the non-optimized cutting process for the full range of the tilted angle δ. Surface roughness parameters were below their upper limit values (Ra ≤ 0.63 μm and Rz ≤ 3.2 μm) regardless of the surface shape and its tilted angle.
In the literature, curvilinear surfaces are more often machined using a milling process, so the proposed optimization method can increase the applicability of the turning process to create the curvilinear surfaces with acceptable and aligned surface roughness parameters.
In the future, the authors plan a modification of the proposed method to apply the new calculation method based on a neural network. The approach should make it possible to provide a more accurate prediction of surface roughness parameters and the impact of additional cutting data such as cutting depth, radius of surface curvature, tool wear and total cutting force components.