In order to fully understand how the factors studied in this paper and their interactions affect the output parameters, statistical analysis is a very useful tool. This section discusses the analysis of variance (ANOVA) to understand the contribution of each factor. The Taguchi statistical analysis of the experimental results has been conducted to complement the graphical analysis and evaluate the contribution of the studied factors, individually and in combination, to the depths induced by the laser hardening and to the maximum hardness reached on the surface. It is also desired to determine the optimal process parameters and to propose models for estimating the depths of the hardened zones and the maximum hardness. The optimal condition has been determined by observing the main effects of each factor, which is a good indicator of the overall impact of the factors [26]. The analysis of variance was performed on MINITAB to determine the regression coefficients and to quantify the contributions of the main effects and the interaction effects between the tested parameters [27]. Tables 5, 6, 7, and 8 summarize the relevant statistical indicators from the variance study for the maximum hardness achieved and the hardening depths Z1 Z2 and Z3. In the present ANOVA study, the stepwise method was used excluding the non-significant terms after each iteration without requiring a hierarchical model. A 95% confidence interval was considered in the statistical calculations. It can be seen that the three parameters considered in the study have an effect either individually or in combination with another factor, the effects of interactions involving three or more factors were omitted from the analysis.
3.4.1 Effect of parameters on the maximum hardness
The main effects graphs are used to interpret the impact of each parameter level on a given average response. A straight line connecting each two levels reflects the influence of each factor level increase on the measured responses. The steeper the slope between the two levels, the more the level of the factor in question affects the response and vice versa. Fig. 8 show the curves of variation of the maximum hardness according to the main effects related to the experimental parameters LP, FS, and RS. The significance of a factor is determined by the change in response to the change in the level of the factor, the greater the change in response the faster the change from a low to a high level of the factor and the steeper the slope in the main effects plot [28, 29]. It can be seen from Fig. 8 that the lines connecting the three levels of each factor are not horizontal, so it can be concluded that the three factors studied have an impact on the maximum hardness but with different degrees. The transition from level 1 to level 2 for LP and FS seems to have a significant effect on the maximum hardness, the slope of the line is very steep in both cases. For RS, the transition from level 1 to level 2 is done with a rather less steep slope, which induces a small change in the maximum hardness. The change from level 2 to level 3 for LP seems to have an opposite effect than the first change from level 1 to level 2. The increase of LP in this phase contributes to the decrease of the maximum hardness, which can be explained by the recurring after melting and material evaporation lead to reduced hardness. In the second pass, FS rises from level 2 to level 3, and the slope is slightly less than that in the first pass. However, within the scope of the study, increasing the feed speed still has a great impact and helps to improve the maximum hardness. For the rotation speed, the slope of the second pass is very small, and the influence on the maximum hardness is relatively small. Table 5 summarizes the variance analysis for the maximum hardness.
According to Table 5, the feed speed (FS) is the most important factor, accounting for 53.5% of the observed hardness. Then, a contribution of 21.04% by the interaction of the sample rotational speed (RS) with the feed speed (FS), while the individual contribution of the rotational speed (RS) is irrelevant. The last important factor is laser power parameter (LP), which contributes with 11.08%. The two remaining factors in the model, which account for 12.11%, represent the interactions of laser power (LP) with scanning speed (FS) and laser power (LP) with sample rotating speed (RS).
Table 5
Maximum hardness variance analysis.
Source
|
Degree
of Freedom
|
Sum of
Squares
|
Contribution
(%)
|
Mean
Squares
|
\({\mathbf{F}}_{0}\)
|
P-value
|
Main effects
|
LP
|
1
|
7,260
|
11,08
|
7,260
|
55,17
|
0,018
|
FS
|
1
|
35,042
|
53,50
|
35,042
|
266,30
|
0,004
|
RS
|
1
|
1,215
|
1,86
|
1,215
|
9,23
|
0,093
|
Two-factors
Interactions
|
LP\(\bullet\)FS
|
1
|
3,721
|
5,68
|
3,721
|
28,28
|
0,034
|
FS\(\bullet\)RS
|
1
|
13,783
|
21,04
|
13,783
|
104,74
|
0,009
|
LP\(\bullet\)RS
|
1
|
4,212
|
6,43
|
4,212
|
32,01
|
0,030
|
|
Error
|
2
|
0,263
|
0,40
|
0,132
|
|
|
|
Total
|
8
|
65,496
|
100
|
|
|
|
R-Squared = % 99.6
|
R-Squared (Adj) = % 98.39
|
Our observations can be confirmed by the P-value in Table 5. Factors with a P value less than 5% are important variables and make significant contributions to the statistical model. Similarly, by comparing the F0 value of each variable with the critical F value taken from the percentage point tables of the F distribution [29], the critical F value can be obtained using the Eq. 7. The results show that all model parameters influence the maximum hardness, except the rotational speed (RS) which has no significant effect on the maximum hardness. Fig. 9 is a graphical representation of Eq. 8, it expresses the evolution on a surface curve of the maximum hardness as a function of the feed rate and the laser power in (a), as a function of the rotation speed and the laser power in (b) and as a function of the feed rate and the rotation speed (c). These curves are very useful from the point of view of optimization and choice of optimal parameters in order to obtain the desired hardness.
It can be seen that the maximum hardness is obtained by increasing the heat input, the latter increases with the increase of the laser power and is inversely proportional to the scan speed. Remember that the scan speed is composed of the rotation speed and the feed speed. On the curve in figure 9 (a), the greater the feed speed, the more important the hardness is, because the function of the feed speed is to move the laser beam from a cross section of the specimen to another.
On the curve of the graph 9 (b), the value of the hardness increases with the increase of the laser power and decreases with the increase of the rotation speed, which is due to the phenomenon of overlapping which increases with the increase of the rotation speed.
On the curve of the graph 9 (c), the maximum value of hardness can be read when the feed speed is maximum and the workpiece rotational speed is minimum. On the other hand, when the feed speed is small and the rotational speed is high, the percentage of overlap of the laser lines will increase which may have the opposite effect on hardening by tempering the previously treated area.
\({F}_{c}={F}_{\alpha ,{DDL}_{1},{DDL}_{2}}= {F}_{\text{0.05,1},2}=18.51\) (Eq. 7)
\({HRC}_{ max}=\text{27,94}+ \text{2,312}\times FS + \text{10,23}\bullet {10}^{-3}\times RS+ \left(1071\times FS - 460\right) {10}^{-6}\times LP-\left(957\times FS+2\times LP\right)\bullet {10}^{-6}\times VR\) (Eq. 8)
3.4.2 Effect of parameters on the hardened zones depth
Figure 10 shows the effects of the factors (LP), (FS), and (RS) on the depth Z1 shown in blue color, Z2 shown in purple color, and Z3 shown in red color. At first sight, it has been found that the lines connecting the three levels of each factor are not horizontal, which means that the three factors studied have an impact on the depths studied but with different degrees. The transition from level (1) to level (2) for (LP) occurred with a less steep slope for the Z1 depth, so it can be said that the effect of power on Z1 in this interval fringe is not very significant. In the case of Z2, the increase of (LP) in this area caused a considerable increase of 74% on Z2. While this increase has been done with a pronounced slope for Z3, causing a considerable increase of 52.31%. The transition from level (2) to level (3) for (LP) is 15 times steeper than the first transition for Z1, so the effect of the increase in (LP) is 15 times steeper when moving to the third level. For the depth Z2, an increase of 4.99% on Z2 has been generated by the increase of (LP) in this scale. For the depth Z3, the passage was made with a medium and negative slope, a decrease of 18,18% on Z3 has been generate by the increase of (LP) in this range.
The transition from level (1) to level 2) of factor (FS) occured with a steep and negative slope, increasing (FS) in this interval has both a large and inverse effect on the depth Z1. Concerning the depth Z2, the passage was made with a more important and negative slope, a fall of 72% of the depth Z2 has been engendered by the increase of (FS) in this first passage. The passage was made with an even more important and negative slope, the increase of (FS) in this first passage caused a fall of 70,27% of the depth Z3. The increase of (FS) at level (3) was done with a positive and important slope, the depth Z1 increased with the increase of (FS) in this case. For the depth Z2, the passage was made with an important and positive slope, the increase of (FS) in this zone leads to a gain of 41,66% of the depth Z2. The passage was made with a moderately large and positive slope, leading to a gain of 27.45% on the Z3 depth.
For (RS), the transition from level (1) to level (2) was rather less steep, the effect of (RS) on Z1 in this interval is minimal. This passage was made with a moderate slope generating an increase of 26,6% on the depth Z2. Concerning the depth Z3, this passage was made with a moderate slope generating an increase of 10,34%. The transition from level (2) to level (3) for (RS) seems to have less effect than the first transition, which was done with a greater three-half slope for the Z1 depth. Similarly, for depth Z2, the passage was made with a moderate slope generating an increase of 16.66%. Contrary to depths Z1 and Z2, depth Z3 seems to be affected by this transition with a significant and negative slope, causing a drop of 41.46%.
3.4.2.1 Effect of parameters on Z1 depth
Table 6 summarizes the descriptive statistical parameters of the variance analysis performed on the hardening depth Z1; we can see that the laser power (LP) has the highest contribution of 44.65%. Sweep speed (FS) appears to have no significant effect on Z1 depth, while rotational speed (RS) has a small contribution of 4.36%. The total contribution of two-factor interactions is estimated at 50.79%. The individual effect of scan speed (FS) is insignificant, but its interaction with laser power (LP) or rotation speed (RS) seems to affect the curing depth. The P-values in the Table 6 are all less than 5% except for the P-value of the scan speed factor (FS). The same observation is made for the value of F0, all the values of F0 are higher than the critical F value except that of the scan speed (FS).
Table 6
Analysis of variance of the hardened depth Z1.
Source
|
Degree
Of Freedom
|
Sum of
Squares
|
Contribution
(%)
|
Mean
Squares
|
F0
|
P-value
|
Main effects
|
LP
|
1
|
106667
|
44,65
|
106667
|
1792,00
|
0,015
|
FS
|
1
|
417
|
0,17
|
417
|
7,00
|
0,230
|
RS
|
1
|
10417
|
4,36
|
10417
|
175,00
|
0,048
|
Two-factors
interactions
|
LP\(\bullet\)LP
|
1
|
27222
|
11,40
|
27222
|
457,33
|
0,030
|
\(\text{L}\text{P}\bullet \text{F}\)S
|
1
|
30250
|
12,66
|
30250
|
508,20
|
0,028
|
\(\text{F}\)S\(\bullet \text{R}\text{S}\)
|
1
|
32111
|
13,44
|
32111
|
539,47
|
0,027
|
LP\(\bullet\)RS
|
1
|
31746
|
13,29
|
31746
|
533,33
|
0,028
|
|
Error
|
2
|
60
|
0,02
|
60
|
|
|
|
Total
|
8
|
238889
|
100
|
|
|
|
R-Squared = % 99.98
|
R-Squared (Adj) = % 99.80
|
For a confidence level α = 5% one can find the critical value of Fc from the percentage point tables of the F distribution [29] according to Eq. 5, \({F}_{\text{0.05,1},1}=161.4\) .
It can be seen that all the parameters of the model represented by Eq. 9, have an effect on the hardening depth except the feed speed (FS) which has no significant effect.
\({Z}_{1}\left(\mu m\right)=3049-\text{2,363}\times LP-\text{31,5}\times FS-\text{0,4107}\times RS+\text{0,02381}\times LP\times FS+ \left(417\times LP\times LP+6250\times FS\times RS+190 LP\times RS\right)\bullet {10}^{-6}\) (Eq. 9)
Figure 11 is a graphical representation of Eq. 9, it expresses the evolution on a surface curve of the hardened region depth Z1 as a function of the feed speed and the laser power in (a), as a function of the rotation speed and the laser power in (b) and as a function of the feed speed and the rotation speed (c). In order to obtain higher values of the depth Z1, the laser power, the feed speed and the rotation speed of the workpiece, must be set to the maximum in their studied intervals.
3.4.2.2 Effect of parameters on Z2 depth
For the depth Z2 representing the hardness loss zone, it can be seen from Table 7 that the model contains six important terms. The interaction coefficient between scanning speed (FS) and rotating speed (RS) contributes the most, accounting for 29.62%, followed by laser power, accounting for 25.19%. The third and fourth positions are scanning speed factor and second-order interaction of scanning speed factor (FS × FS), and their contribution rates are 18.92% and 19.03% respectively. In the depth model of Z2, the rotational speed factor (RS) contributes 5.49%, and the interaction factor between laser power and rotational speed (RS) contributes 1.69%. The P-values in Table 7 are less than 5%, which indicates that all variables are important in the statistical model and have significant contributions.
Table 7
Variance analysis of hardening depth Z2.
Source
|
Degree
Of Freedom
|
Sum of
Squares
|
Contribution
(%)
|
Mean
Squares
|
F0
|
P-value
|
Main effects
|
LP
|
1
|
93750
|
25,19
|
93750
|
771,43
|
0,001
|
FS
|
1
|
70417
|
18,92
|
70417
|
579,43
|
0,002
|
RS
|
1
|
20417
|
5,49
|
20417
|
168,00
|
0,006
|
Two-factors
interactions
|
\(\text{F}\)S\(\bullet \text{R}\text{S}\)
|
1
|
110250
|
29,62
|
110250
|
907,20
|
0,001
|
LP\(\bullet\)RS
|
1
|
6298
|
1,69
|
6298
|
51,82
|
0,019
|
\(\text{F}\)S\(\bullet\)FS
|
1
|
70848
|
19,03
|
70848
|
582,98
|
0,002
|
|
Error
|
2
|
243
|
0,07
|
122
|
|
|
|
Total
|
8
|
372222
|
100
|
|
|
|
R-Squared = % 99.93
|
R-Squared (Adj) = % 99.74
|
This can be confirmed by comparing the F0 value of each variable with the critical FC value in the percentage table of distribution F [29]. For the confidence level α = 5%, the critical FC value can be found according to Eq. 7, F0.05,1,2 = 18.51.
The model obtained by the statistical analysis for the depth of hardness loss zone Z2 is formulated by Eq. 10 and represented graphically in Fig. 12. It can be clearly seen that the maximum on depth Z2 can be obtained when the laser power is high enough between 2400 W and 2600 W, and the feed and rotation speeds are set to their minimum and maximum respectively. We can therefore deduce that the depth Z2 increases with the increase in the amount of the heat received by the treated workpiece.
\({Z}_{2}\left(\mu m\right)=\text{500,3}+\text{0,2125}\times LP-\text{483,9}\times FS+\text{0,4735}\times RS+\text{53,91}\bullet (FS\times FS)-48\times {10}^{-6} (LP\times RS)-\text{0,05729}\bullet (FS\times RS)\) (Eq. 10)
3.4.2.3. Effect of parameters on depth Z3
For the overtempered zone depth specified by Z3, it can be seen from Table 8 that the model includes six important parameters and seven degrees of freedom for the considered confidence interval. The laser power (LP) contributes 20.53%, as the factors rotational speed (RS) and scanning speed (FS) were considered insignificant and were excluded from the model. The interaction factor between laser power and rotation speed (LP × RS) has the largest contribution of 38.19%. The second-order interaction factor of the laser power (LP × LP) contributes with 23.00%, the remaining interaction factors having an effect in the model of the depth of the Z3 superheat zone participate with small contributions included in the interval (1.71 – 7.35), for an overall contribution totaling 18.27%.
Table 8
Variance analysis of the hardened depth Z3
Source
|
Degree
Of Freedom
|
Sum of
Squares
|
Contribution
(%)
|
Mean
Squares
|
F0
|
P-value
|
Main effects
|
LP
|
1
|
240000
|
20,53
|
131026
|
738,60
|
0,023
|
Two-factors
Interactions
|
LP\(\bullet\)LP
|
1
|
268889
|
23,00
|
161364
|
909,62
|
0,021
|
\(\text{F}\)S\(\bullet\)FS
|
1
|
39470
|
3,38
|
172814
|
974,16
|
0,020
|
\(\text{R}\)S\(\bullet\)RS
|
1
|
68098
|
5,83
|
124148
|
699,83
|
0,024
|
LP\(\bullet\)FS
|
1
|
85934
|
7,35
|
68388
|
385,50
|
0,032
|
LP\(\bullet\)RS
|
1
|
446348
|
38,19
|
410900
|
2316,26
|
0,013
|
\(\text{F}\)S\(\bullet \text{R}\text{S}\)
|
1
|
19973
|
1,71
|
19973
|
112,59
|
0,060
|
|
Error
|
1
|
177
|
0,02
|
177
|
|
|
|
Total
|
8
|
1168889
|
100
|
|
|
|
R-Squared = % 99.98
|
R-Squared (Adj) = % 99.88
|
The P-values in Table 8 are all less than 5%, which implies that all variables are significant and contribute significantly to the statistical model, except for the interaction (FS∙RS) where the P-value is 6%. This remark can be confirmed by comparing the F0 values of each variable with the critical Fc value taken from the percentage point tables of the F distribution [29], for a confidence level α = 5% the critical value of Fc (Fc = F0.05,1,1 = 161.4) was calculated according to Eq. 7.
The model obtained by the statistical analysis for the depth of over-tempering zone Z3 is expressed by Eq. 11 and represented graphically in Fig. 13. it can be seen that the depth Z3 is maximized by average values of the laser power around (2100 – 2600 W), feed speed at minimum and maximum values and average values of the rotation speed. The graph (c) of Fig. 13, tells us that there are two combinations to maximize the Z3 value. When the laser power is set to 2100 W, the feed speed must be at its maximum and the rotation speed must be at its minimum, and when the laser power is set to 2600 W, the feed speed must be at its minimum and the rotation speed must be at its maximum. It can be understood that the depth Z3 is proportional to the interaction time between the laser beam and the material.
\({Z}_{3}\left(\mu m\right)=-4918+\text{5,560}\times LP-\text{0,001268} \left(LP\times LP\right)+\text{43,90} \left(FS\times FS\right)-\left(73 \left(RS\times RS\right)+304 \left(LP\times RS\right)\right){10 }^{-6}-\text{0,19120} (LP\times FS)-\text{0,02479}\bullet (FS\times RS)\) (Eq. 11)
The proposed statistical models demonstrate excellent reproducibility of the experimental results, with a maximum error of 2.66% for the maximum hardness approximation model. We also report very low maximum errors on the hardness depth approximation models, with errors of 5.748%, 5.113%, and 2.68% for depths Z1, Z2, and Z3, respectively.