3.1. Results and discussion for parameter Ra
From the viewpoint of the chip machining of the coating after it’s spraying with the alloy Stellite 6, two specific problems arose that caused worse machinability. From the viewpoint of machinability, it was mainly the boundaries of the splat, which did not concern the grain boundaries but the boundaries of the deformed particles. Cohesiveness between splats is lower, particularly for Stellite 6. The splat boundaries can be formed by the oxides formed during the flame particle flight during flame annealing. The second problem was the alloy material as such. The Stellite within the splat undergoes a deformation induced martensite transformation from the fcc to the hcp grid. During machining, it hardens and becomes brittle, similarly as some types of steel. The behaviour of the investigated objects can generally be described using models that can explain the logical relationships of objects, but sometimes they must do with finding a mathematical function that determines the dependence between factors and responses for the objects in question, although it is impossible to answer unequivocally why is it so. The monitored responses in the presented experiment can be searched and verified using a mathematical model using statistical tools. This method makes it possible to describe how the given response depends on individual factors. Due to the fact that, in the case of surface roughness profile parameters the attention was paid mainly to the basic parameters of roughness Ra, Rz, two models were sought for these two responses. For the Ra parameter, the model is formulated in (1) as a relation with variables vc cutting rate, frev feed rate and aP depth of cut.

The analysis shows that the model (1) with the action of the constant tool RNGN43 LX11 TUNGALOY accounts for 99.81% of the variability of the investigated parameter Ra. This conclusion can be accepted on the basis of the value of the adjusted index of determination. The mean value of Ra = 1.98 μm and estimation of the standard deviation of random error is 0.051968. The chosen mathematical and statistical dependence according to the Cohen scale can, therefore, be considered as functional. Furthermore, the adequacy of the selected model was tested using analysis of variance (Table 5) by testing the zero-statistic hypothesis H0, which resulted from the nature of the test and provided information that none of the effects used in the model affected the significant change of the investigated variable. It followed from the subject test that the achieved level of significance (Prob> F) was less than the chosen significance level α = 0.05 and that it could be concluded that there was not enough evidence for accepting the H0 hypothesis, i.e. that it could be stated that the model (1) was significant. It also follows that part of the total variability of the experimentally obtained values, which corresponds to random errors, is significantly smaller than the variability of the measured values in accordance with the model.
[Table 5 about here.]
Table 5 Analysis of variance by testing the zero statistic hypothesis H0.
Source
|
DF
|
Sum of Squares
|
Mean Square
|
F Ratio
|
Prob > F
|
Model
|
3
|
1.2680028
|
0.422668
|
2710.275
|
<0.0001
|
Error
|
12
|
0.0018714
|
0.000156
|
|
|
C. Total
|
15
|
1.2698742
|
|
|
|
On the basis of testing of the adequacy of the used statistical model lzr it is possible to determine the regression coefficients of statistical dependence defined in (1) using the smallest square method (Table 6).
[Table 6 about here.]
Table 6 Estimates of the parameter of the investigated dependence for the variable Ra.
Term
|
|
Estimate
|
Std Error
|
t Ratio
|
Prob>|t|
|
Lower 95 %
|
Upper 95%
|
Const
|
|
-1.7870540
|
0.137774
|
-12.97
|
<0.0001*
|
-2.0872390
|
-1.4868700
|
x1
|
|
0.5748062
|
0.039803
|
14.44
|
<0.0001*
|
0.4880820
|
0.6615305
|
x2
|
|
2.4733928
|
0.050146
|
49.32
|
<0.0001*
|
2.3641331
|
2.5826525
|
x3
|
|
-1.4709540
|
0.098357
|
-14.96
|
<0.0001*
|
-1.6852560
|
-1.2566530
|
* - significant at the significance level α = 0.05.
After substitution of the values of the individual regression coefficients (Table 6) calculated by the least squares method into the general form of the model (1) we get the final form of the dependence (2) for further analysis of the dependence of Ra value on the cutting conditions.

An analysis of the influence of the individual investigated input variables on the change of the Ra value is shown in Figures 2 and 3. It is obvious that with an increase in the cutting speed the value of the mean arithmetic deviation of the roughness profile also increases. Figure 2 illustrates an experimentally obtained model describing the given relationship for constant parameters: constant feed rate, constant cutting depth, and constant machining length, due to the influence of the given input factors on the change of the Ra value. The influence of the cutting rate on the change of observed response Ra is 15.75%. Given the validity of the predictive model (1) at the selected range of the cutting rate from 150 to 250 m·min-1, it can be stated that within the given interval an increase of the cutting rate by 10 m·min-1 will cause an increase of the Ra value by 0.027 μm, with fulfilment of other conditions ensuing from the nature of mathematical and statistical modelling. The validity of the model (1) is limited to the interval of the used cutting conditions, and the validity beyond of those intervals has to be always verified experimentally.
[Fig. 2 about here.]
The feed rate frev as the second investigated input factor has a 53.79 % influence on the change in the value of the observed response of the surface roughness Ra. It may be stated that with an increase in the feed value the value of the mean arithmetic deviation of the roughness profile increases as well. The influence of feed rate on the roughness of the machined surface results from the kinematics of the machining technology. From the machining theory, it is obvious that with the increase in feed rate the corrugation of the machined surface increases. Research performed by many authors shows that, at higher feed rates, the cutting rates have a smaller influence on the quantitative values describing the micro-geometry of the machined surface. These conclusions are also documented by the results of this experiment (Figure 3).
[Fig. 3 about here.]
More profound analysis of the statistically most significant factor - the feed rate frev on the change of the observed response Ra, shows that the conditional value Ra (Figure 3) also increases within the range of the experimentally used values of feed rate with its increase. Since the feed rate plays a major role in changing the Ra value, its influence can be further analysed by the Kruskal-Wallis nonparametric test (Table 7). The Kruskal-Wallis analysis of variance tests the zero hypothesis, i.e. that all mean values in the investigated groups, in this case in the individual levels of experimentally used feed rates, are the same. On the basis of the achieved level of significance of the Kruskal-Wallis test (p = 0.017), it can be concluded that at the chosen level of significance α = 5 % a statistically significant difference in the value of the mean arithmetic variance of the roughness profile Ra based on the change of the feed rate value was proven. This conclusion can be naturally accepted only on the basis of an assumption that the cutting rate and the depth of cut do not affect the change of the Ra value. However, since their impact is within the interval of 15.75 % for the cutting rate and 16.32 % for the cutting depth, this assumption can be accepted for the discussed need.
[Table 7 about here.]
Table 7 Results of the Kruskal-Wallis test of difference of the Ra value in dependence on the change of the feed rate.

It is necessary to realise in this context that the Kruskal-Wallis test does not work with the original values but with the sequential numbers that were assigned. From the values of the sum of the order (Figure 8), it can be seen that the lowest values of the observed response Ra were achieved at the feed rate of 0.4 mm/hr-1 and with the continuous increase of the feed rate value the Ra values also increased. The results are consistent with the previous conclusions. When comparing the achieved levels of significance p (Table 8), it can be seen that the statistically significant difference in the achieved Ra value is only between the feed rate of 0.4 mm·rev-1 and 1.0 mm·rev-1. The difference of the observed response Ra between the mm·rev-1 and 0.6 mm·rev-1, as well as between 0.6 mm·rev-1 and 1.0 mm·rev-1 was not proven.
[Table 8 about here.]
Table 8 Multiple comparisons of the p-values of the differences of Ra depending on the change in the feed rate frev.

The third independently examined variable factor was the cutting depth aP. It is affected by 16.32 % in the case of a change in the investigated Ra response. The influence of the cutting depth on the surface roughness results from changes in the deformation processes in the cutting zone when changing the depth of cut. The literature sources document a negligible influence of the cutting depth on the roughness of the surface. By increasing the cutting depth, the roughness will decrease slightly if the cutting rate and feed rate are constant.
3.2. Results and discussion for parameter Rz
Similarly, as for the investigated parameter Ra, it is also possible for the investigated parameter Rz to analyse the assumed dependences and to express the model (3)

An Analysis of the investigated parameter Rz shows that the model (3) under a constant action of the tool RNGN43-LX11-TUNGALOY represents 97.22 % of the variability of the investigated parameter Rz at the changes of the cutting conditions. This conclusion can be accepted on the basis of the value of the adjusted index of determination. The average value Rz = 10.546 μm and estimate of the standard deviation of random error are 0.83876. The chosen mathematical and statistical dependence according to the Cohen scale can, therefore, be considered functional. The analysis of the model (3) using the Fisher's analysis of variance points to the fact that from the total variability of the model with 15 degrees of freedom and the sum of squares of deviations of 1.0729, only 0.02389 of the sum of squares of deviations represent errors, which is 2.23 %. The remaining almost 98 % of the total variability belongs to the model. On the basis of the achieved level of significance of the Fisher-Snedecor test criterion (p = 0.0001), it can be concluded that the model (3) contains at least one effect that significantly affects the change in the Rz value. Therefore, at the chosen level of significance, the model (3) can be considered adequate. On the basis of the above analysis, it is possible to obtain individual coefficients of equation (3) using the least squares method with the identification of their statistical significance.
[Table 9 about here.]
Table 9 Estimates of the parameter of the investigated dependence for the variable Rz.
Term
|
Estimate
|
Std Error
|
t Ratio
|
Prob>|t|
|
Lower 95 %
|
Upper 95 %
|
const
|
-0.9965750
|
0.492307
|
-2.02
|
0.0658
|
-2.069220
|
0.076070
|
x1
|
0.2392116
|
0.142229
|
1.68
|
0.1184
|
-0.070680
|
0.549102
|
x2
|
2.5029316
|
0.179188
|
13.97
|
<0.0001*
|
2.112515
|
2.893348
|
x3
|
-2.2118640
|
0.351458
|
-6.29
|
<0.0001*
|
-2.977630
|
-1.446100
|
It follows from Table 9 that a statistically significant factor influencing the change of the investigated response Rz is primarily the feed rate, with a 58.305 % influence on its change, as well as the cutting depth that contributes to a change in Rz variability of 26.252 %. For the parameter Rz, at the level of significance of α = 5 %, no significant influence of the cutting rate and an absolute element, or of the model constant (3) was proven. Precisely the statistical insignificance of the constant indicates that within the realised experiment only important input factors were used. By substituting the coefficients from Table 9 to the model (3), we obtain the final form of the statistical prediction function of the dependence of the value of the investigated response Rz on the cutting conditions for the feed turning of the coating Stellite 6 with the use of the tool RNGN43 LX11-TUNGALOY (4). These conclusions are also confirmed by simulation of experimentally obtained results using the neural network. At the nodes of a neural network, analogously called neurons, each input variable xi at the input of the jth neuron is multiplied by the weighting factor wji. The sum zj = w0j + Σwji·xi is in a neuron transformed by the activation function. The activation function expresses the intensity of the neuron response to the given input. The most commonly used activation functions include the logistic function, σj(z)=1/(1+e-z), which resembles the biological sensoric response function. The weights wji represent the intensity of the linkage between the variable and the neuron, or, in the case of multilayer networks, between the neurons in the layers. These linkages are sometimes called synapses. The output variables are predicted as weighted linear combinations of outputs from the last hidden neuron layer. Thus, the neural network is formally a special case of multiple nonlinear regression; the neural network can be virtually considered to be a nonparametric regression. If the neural network contains no hidden layer of neurons, only the input and output variables, it would be a linear regression model. The neural network is optimised using the smallest square criterion. This means that the network is set in a way that the sum of squares of the predicted and measured value of the output variable is minimal. This setting is the objective of an iterative optimisation procedure, called learning or training of a neural network. The optimisation procedure uses adaptive Gauss-Newton algorithms. For the application of the neural network, the experimentally obtained response values Rz were divided by a ratio of 85 % to 15 %, the first group being used as data for learning and the second group as data for testing. The basic used neural network with three layers (the first containing 3 neurons and the other two hidden layers contained 2 neurons each) reached a maximum error for learning data value of 0.004346201313 and a mean error for learning data value of 0.0009920144294. The maximum error for the tested data is 0.003247050035 and the mean error is 0.001291113035. These values, along with the graphical representation of the learning process (Figure 4), demonstrate good quality of the model and they express both the improvement of the model and the improvement of the prediction of valid data. The model in question therefore predicts well from unknown data, and at the same time, it predicts both curves, i.e. both for learning data and for testing data.
[Fig. 4 about here.]
From the analysis of the model generated by the neural network, the residues represent 12.026 from the total sum of the deviations squares of 768.408, which is 1.57 %. The remaining 98.44 % represents an explanation of the sum of squares by the model. The achieved level of significance of the Fisher test criterion reaches the value p = 0.00417, which means that at the significance level α = 5 % the model can be considered to be significant. From the viewpoint of the relative influence of the predictors on the change in the value of the monitored response Rz (Figure 5), it can be concluded that it is namely the feed rate, which has the dominant influence. The cutting depth also contributes significantly to the change in the value of the Rz parameter and, as in previous analyses, the influence of the cutting speed was only minimal.
The influence of the feed rate (Figures 5 and 6) on the change of the Rz (4) value is connected not only with the geometric causes of formation of the resulting surface micro-geometry, but it also considerably influences the elastic and plastic deformations in the surface layer.

[Fig. 5 about here.]
[Fig. 6 about here.]
The influence of the feed rate on the change of the Rz value has to be also observed in relation to the cutting rate. If we increase the feed rate from 0.4 mm·rev-1 to 0.6 mm·rev-1 at a constant cutting rate and constant cutting depth, the value of the parameter Rz will increase by 175.90 %, i.e. from the value of 5.5 μm to the value of 15.175 μm. If at a constant cutting rate and a constant cutting depth we increase the feed rate from the value of 0.6 mm·rev-1 to 1.0 mm·rev-1, the value of the Rz parameter will increase already by 259.16 %. It means therefore that at maintained constant cutting conditions (cutting rate, cutting depth), when the feed rate increases from 0.4 mm·rev-1 to 0.5 mm·rev-1, then the value of the roughness parameter Rz will increase by 74.81 %. This increase in roughness value Rz decreases with an increasing feed rate. The average value of the change in the roughness Rz at an increase in feed rate by 0.1 mm·rev-1 represents the value of 47.312 mm·rev-1.
When observing the change in the Rz value in dependence on the change in feed rate with simultaneous change in the cutting rate (Figure 7), it can be observed that by increasing the cutting rate, the Rz value increases with the increased feed rate, while the change in the Rz value caused by the cutting rate is small. It varies from 2.55 % to 3.75 % with an increase of the cutting rate from 150 m·min-1 to 250 m·min-1 in increments of 25 m·min-1. The cutting depth has a more significant influence on the change of the Rz value. An increase of the cutting depth from 0.1 mm to 0.15 mm reduces the Rz value by approx. 59 %. However, this reduction in the given cutting depth interval must be attributed to the machined material as such. When increasing the cutting depth by more than 0.15 mm, it is possible to expect on the contrary a deterioration of the roughness Rz.
[Fig. 7 about here.]