Analysis and Optimization of Bead Geometry by Using Response Surface Methodology

Abstract

Analysis of bead geometry is very important in product design and manufacturing. Defect-free products with reliability are the demanding parameter in the manufacturing Industry. In this study, we have analyzed and optimized bead geometry parameters such as height of reinforcement (HOR), depth of Heat Affected Zone (DOH), and width of Heat Affected Zone (WOH) by using Central Composite Design (CCD) of response surface methodology (RSM). In this study, peak current and pulse frequency are the most important process parameters for HOR and the optimum combination obtained are (160 A, 80 A, 100 Hz, and 45%) further HOR at this optimum was found to be 1.41 mm, which is close to 1.45 mm. Similarly, peak current and pulse frequency are the most important process parameter for WOH and the optimum combination obtained are (160 A, 80 A, 150 Hz, and 45%) further WOH at this optimum was found to be 1.32 mm, which is close to 1.37 mm. Again, similarly peak current and pulse frequency are the most important process parameter for DOH and the optimum combination obtained are (160 A, 80 A, 100 Hz, and 45%) further DOH at this optimum was found to be 1.26 mm which is close to 1.58 mm.

1. Introduction

The traditional method of selecting one parameter is time taking process and therefore not considered nowadays in the manufacturing industry, hence an optimization technique concerning the design of experiment (DOE) such as CCD of response surface methodology (RSM) to establish an optimum condition for tensile strength. In this study, the surface plot is used to explain the main and interaction effect of the process parameter to identify the optimum parameter with their values. RSM is a widely used statistical technique in process optimization [1]. RSM is a set of mathematical and statistical methodologies for assessing problems in which multiple independent factors influence a dependent variable or response, to optimize the answer. RSM facilitates the examination of the interaction between experiment variables within the range under consideration, allowing for a better knowledge of the process while lowering experiment time and cost [2, 3].

2. Steps of Response Surface Methodology

Major steps of RSM are shown in Figure 1.

Figure 1.

Flowchart representing steps of RSM.

2.1 Input parameters and their operating range

Based on a review of the literature and previous research, the most important process parameters that have a greater influence on bead geometry and mechanical properties have been identified. The butt joint was made from AISI 316 stainless steel sheets with dimensions of 100 × 75 × 4 mm by used pulsed TIG welding [4]. This experiment's input parameters are peak current, base current, pulse frequency, and pulse on time [2]. Input parameters with their levels are given in Table 1. The experiment was carried out at an optimum in the laboratory.

Input parameter	Factor symbol	Level 1	Level 2	Level 3	Level 4	Level 5
		−$\beta$	−γ	$\alpha$	$\beta$	γ
Peak current (I)	A	140	150	160	170	180
Base current (I)	B	60	70	80	90	100
Pulse frequency (Hz)	C	50	75	100	125	150
Pulse on time (%)	D	35	40	45	50	55

Table 1.

Independent parameters with their levels for CCD.

2.2 Design of experiment

The experimental design for this investigation is CCD and the response is measured by RSM. Examine the combined effect of four different input parameters on bead geometry and mechanical properties to optimize the process parameter of pulse TIG welding and drive a mathematical model. Five levels, four-parameter CCD which include 24 = 16 factorial points plus 6 central points and 2 × 4-star points (24 + 2*4 + 6) [2, 5], with a total of 30 experiments were made in this investigation as shown in Table 2. The framework for the four factors ranged between five levels, −γ, α, +$\beta$, and +γ).

Std	Factor symbol				Actual factor
	A	B	C	D	A	B	C	D
1	−γ	−γ	−γ	−γ	150	70	75	40
2	β	−γ	−γ	−γ	170	70	75	40
3	−γ	β	−γ	−γ	150	90	75	40
4	β	β	−γ	−γ	170	90	75	40
5	−γ	−γ	β	−γ	150	70	125	40
6	β	−γ	β	−γ	170	70	125	40
7	−γ	β	β	−γ	150	90	125	40
8	β	β	β	−γ	170	90	125	40
9	−γ	−γ	−γ	β	150	70	75	50
10	β	−γ	−γ	β	170	70	75	50
11	−γ	β	−γ	β	150	90	75	50
12	β	β	−γ	β	170	90	75	50
13	−γ	−γ	β	β	150	70	125	50
14	β	−γ	β	β	170	70	125	50
15	−γ	β	β	β	150	90	125	50
16	β	β	β	β	170	90	125	50
17	−β	α	α	α	140	80	100	45
18	α	α	α	α	180	80	100	45
19	α	−β	α	α	160	60	100	45
20	α	α	α	α	160	100	100	45
21	α	α	−β	α	160	80	50	45
22	α	α	α	α	160	80	150	45
23	α	α	α	−β	160	80	100	35
24	α	α	α	α	160	80	100	55
25	α	α	α	α	160	80	100	45
26	α	α	α	α	160	80	100	45
27	α	α	α	α	160	80	100	45
28	α	α	α	α	160	80	100	45
29	α	α	α	α	160	80	100	45
30	α	α	α	α	160	80	100	45

Table 2.

Design of experiment or central composite design arrangement.

3. RSM statistical analysis for reinforcement height

By varying the input process parameter, CCD was used to experiment. The experiment was carried out by varying the input parameters with the experimental design CCD. The experiment was carried out using various parameter combinations, as shown in Table 3. The CCD experiment results were fitted to the polynomial regression equation created by Design Expert Software 18.0 [2, 6].

3.1 Development and evaluation of regression equation HOR

The correlation between process parameters and output response was obtained by using CCD The second-order polynomial regression equation fitted between the output response and the input process parameter. From the ANOVA result shown in Table 4, it has been found adequacy of the model is suitable to analyze the experimental value [2, 6].

Std	Factor symbol				Actual factor				Exp. value; Bead width
	A	B	C	D	A	B	C	D
1	−γ	−γ	−γ	−γ	150	70	75	40	0.85
2	β	−γ	−γ	−γ	170	70	75	40	0.68
3	−γ	β	−γ	−γ	150	90	75	40	0.57
4	β	β	−γ	−γ	170	90	75	40	0.93
5	−γ	−γ	β	−γ	150	70	125	40	0.74
6	β	−γ	β	−γ	170	70	125	40	0.64
7	−γ	β	β	−γ	150	90	125	40	1.01
8	β	β	β	−γ	170	90	125	40	0.76
9	−γ	−γ	−γ	β	150	70	75	50	0.67
10	β	−γ	−γ	β	170	70	75	50	1.05
11	−γ	β	−γ	β	150	90	75	50	0.75
12	β	β	−γ	β	170	90	75	50	0.63
13	−γ	−γ	β	β	150	70	125	50	1.08
14	β	−γ	β	β	170	70	125	50	0.80
15	−γ	β	β	β	150	90	125	50	0.71
16	β	β	β	β	170	90	125	50	1.14
17	−β	α	α	α	140	80	100	45	0.88
18	α	α	α	α	180	80	100	45	0.78
19	α	−β	α	α	160	60	100	45	1.14
20	α	α	α	α	160	100	100	45	0.81
21	α	α	−β	α	160	80	50	45	0.67
22	α	α	α	α	160	80	150	45	1.17
23	α	α	α	−β	160	80	100	35	0.87
24	α	α	α	α	160	80	100	55	0.71
25	α	α	α	α	160	80	100	45	1.31
26	α	α	α	α	160	80	100	45	0.92
27	α	α	α	α	160	80	100	45	0.76
28	α	α	α	α	160	80	100	45	1.07
29	α	α	α	α	160	80	100	45	1.18
30	α	α	α	α	160	80	100	45	1.45

Table 3.

CCD experimental value: HOR.

Source	Coefficient	Sum of squares	df	Mean square	F-values	p-value
Model	1.11	0.7246	14	0.7246	1.10	0.0425	Significant
A	0.0024	0.0003	1	0.0003	3.003	0.047
B	−0.0275	0.0181	1	0.0181	0.087	0.5331
C	0.0691	0.1307	1	0.1307	2.79	0.0115
D	0.0144	0.0044	1	0.0044	0.1028	0.7456
A × B	0.0375	0.0235	1	0.0235	0.4609	0.0498
A × C	−0.0406	0.0276	1	0.0276	0.5849	0.4643
A × D	0.0356	0.0208	1	0.0208	0.4268	0.5107
B × C	0.0444	0.0328	1	0.0328	0.5729	0.4359
B × D	−0.0456	0.0356	1	0.0356	0.7023	0.4123
C × D	0.0325	0.0168	1	0.0168	0.3508	0.5670
A^2	−0.0824	0.1862	1	0.1862	3.98	0.0447
B^2	−0.0461	0.0584	1	0.0584	1.25	0.2817
C^2	−0.0599	0.0984	1	0.0984	2.10	0.1678
D^2	−0.0918	0.2310	1	0.2310	4.93	0.0422
Residual		0.7025	15	0.0468
Lack of fit		0.3837	10	0.0384	0.6017	0.7691	Not significant
Pure error		0.3188	5	0.0638
Cor total	1.11	0.7230	14	0.0516	1.10	0.0425	Significant

Table 4.

ANOVA for the: HOR.

R² = 0.99543, adjusted R² = 0.99763.

The regression equation based on the regression coefficient of ANOVA results is shown in Eq. (1).

$$\mathrm{HOR}=\begin{array}{l}1.11+0.0025\mathrm{A}-0.0275\mathrm{B}+0.0737\mathrm{C}+0.0142\mathrm{D}+0.0375\mathrm{AB}\end{array}$$

$$\begin{array}{l}-0.0406\mathrm{AC}+0.0356\mathrm{AD}+0.0444\mathrm{BC}-0.0456\mathrm{BD}+0.0325\mathrm{CD}\end{array}$$

$$\begin{array}{l}-0.0824{\mathrm{A}}^2-0.0461{\mathrm{B}}^2-0.0599{\mathrm{C}}^2-0.0918{\mathrm{D}}^2\end{array}$$E1

To obtain a statistically significant regression model p-value, if the p-value < 0.05 then the mathematical model is significant. A, C, AB, A², and D² are significant model terms in this case. The model can be reduced to Eq. (2), after eliminating the insignificant coefficients. After that predicted value for all the combinations of input, the parameter is obtained as shown in Table 5.

Std	Factor sign				Estimated value	Remaining error
	A	B	C	D
1	−γ	−γ	−γ	−γ	0.849	0.012
2	β	−γ	−γ	−γ	0.789	−0.098
3	−γ	β	−γ	−γ	0.722	−0.136
4	β	β	−γ	−γ	0.812	0.134
5	−γ	−γ	β	−γ	0.924	−0.168
6	β	−γ	β	−γ	0.702	−0.046
7	−γ	β	β	−γ	0.974	0.047
8	β	β	β	−γ	0.902	−0.126
9	−γ	−γ	−γ	β	0.832	−0.151
10	β	−γ	−γ	β	0.915	0.146
11	−γ	β	−γ	β	0.522	0.239
12	β	β	−γ	β	0.755	−0.109
13	−γ	−γ	β	β	1.037	0.059
14	β	−γ	β	β	0.957	−0.141
15	−γ	β	β	β	0.905	−0.179
16	β	β	β	β	0.975	0.181
17	−β	α	α	α	0.794	0.097
18	α	α	α	α	0.804	−0.013
19	α	−β	α	α	0.999	0.152
20	α	α	α	α	0.889	−0.068
21	α	α	−β	α	0.741	−0.060
22	α	α	α	α	1.036	0.145
23	α	α	α	−β	0.733	0.148
24	α	α	α	α	0.790	−0.064
25	α	α	α	α	1.129	0.193
26	α	α	α	α	1.129	−0.193
27	α	α	α	α	1.129	−0.358
28	α	α	α	α	1.129	−0.043
29	α	α	α	α	1.129	0.068
30	α	α	α	α	1.129	0.333

Table 5.

CCD predicted value: HOR.

$$\mathrm{HOR}=1.11+0.0025\mathrm{A}+0.0737\mathrm{C}+0.0375\mathrm{AB}-0.0824{\mathrm{A}}^2-0.0918{\mathrm{D}}^2$$E2

3.2 Adequacy check of the mathematical model for height of reinforcement

ANOVA represents that the polynomial regression equation was significant to represent the relationship between input parameters and output parameters. The adequacy and significance of the established model were also elaborated by the high value of the coefficient of determination (R²) value of 0.99543 and adjusted R² 0.99763 for the development of the developed correlation [2, 3].

Figure 2 demonstrates that the regression model generated with Design Expert 18.0 has a good correlation between the experimental and predicted values since all of the points are very close to the line of perfect fit or line of unit slope. Furthermore, residuals were investigated to validate the model’s adequacy. The difference between the observed and predicted responses is referred to as the residual. This analysis was examined using the normal probability plot of residuals [2, 5]. The normal probability plot of the residuals shows that the errors are distributed normally in a straight line and are insignificant as shown in Figure 3.

Figure 2.

Plot of experimental vs. predicted value HOR.

Figure 3.

Normal probability plot of residual HOR.

3.3 Perturbation plot: height of reinforcement

The perturbation plot shows the effect of all the parameters on a single plot A perturbation plot to compare the effect of all the process parameters at the center point on bead width is presented in Figure 4. It has been noted that HOR peak current (A) is increasing and then HOR decreases [1, 6].

Figure 4.

Perturbation plot of HOR.

The plot also shows that the HOR decreases as the base current (B) increases because no melting occurs during this stage. This plot shows that HOR increases as pulse frequency (C) increases. The plot also shows that HOR increases as pulse on-time increases (D) and then decreases [2].

3.4 Response surface plot: height of reinforcement

The 3D surface plot and 2D contour effect developed by design expert 18.0 software represent the interaction effect between process parameters and HOR as shown in Figures 5–10 [3].

Figure 5.

Surface plot (a), contour plot (b) of the interaction effect AB on HOR.

Figure 6.

Surface plot (c), contour plot (d) of the interaction effect AC on HOR.

Figure 7.

Surface plot (e), contour plot (f) of the interaction effect AD ion HOR.

Figure 8.

Surface plot (g), contour plot (h) of the interaction effect BC ion HOR.

Figure 9.

Surface plot (i), contour plot (j) of the interaction effect BD on HOR.

Figure 10.

Surface plot (k), contour plot (l) of the interaction effect CD on HOR.

The coefficient of the linear interactive effect of peak current and base current is positive as given in Table 4. HOR is increased as the value of the above parameter is increased as shown in Figure 5a of the 3D surface plot and Figure 5b of the contour plot. HOR increases with concurrent increases in peak current and base current to approximately 180−100 A, respectively, beyond which the value of HOR decreases [2, 3]. As shown in Table 4, the coefficient of linear interactive effects of peak current and pulse frequency is negative. As shown in Figure 6c of 3D surface plots and Figure 6d of contour plots, HOR increases as the value of the above parameter increases. The HOR declined beyond the peak current of 180 A and pulse frequency of 125 Hz respectively [2, 6].

As shown in Table 4, the coefficient of the linear effect of peak current and pulse on time is positive. As shown in Figure 7e of the 3D surface plot and Figure 7f of the contour plot, HOR increases as the value of the above parameter increases. DOP is increasing with simultaneously increasing in peak current and pulse on time to about 180 A and 50% respectively beyond which the value of HOR declined. Table 4 shows that the coefficient of the linear effect of base current and pulse frequency is positive. As shown in Figure 8g of the 3D surface plot and Figure 8h of the contour plot, HOR increases as the value of the above parameter increases. DOP rises as peak current and pulse on time rise to around 180 A and 50%, respectively, beyond which the value of HOR tends to fall. Table 4 shows that the coefficient of linear interactive effects of base current and pulse on time is negative [2]. As shown in Figure 9i of the 3D surface plots and Figure 9j of the contour plot, BW increases as the value of the above parameter increases. Beyond the base current of 100 A, the HOR and pulse on time both decreased by 50%. The coefficient of the linear interactive effect of pulse frequency and pulse on time is positive as given in Table 4. As the value of the above parameter is increased, BW increases, as shown in Figure 10k of the 3D surface plot and Figure 10l of the contour plot [1, 2]. HOR is increasing with simultaneously increasing pulse frequency and pulse on time to about 125 Hz and 50% respectively beyond which the value of HOR declined.

4. Statistical analysis for depth of heat affected zone using RSM

By varying the input process parameter, CCD was used to experiment. The experiment was carried out by varying the input parameters with the experimental design CCD. The experiment was carried out using various parameter combinations, as shown in Table 6. The CCD experiment results were fitted to the polynomial regression equation created by Design Expert Software 18.0 [2, 5].

4.1 Development and evaluation of regression equation: depth of HAZ

The correlation between process parameters and output response was obtained by using CCD. The second-order polynomial regression equation fitted between the output response and input process parameter. From the ANOVA result shown in Table 7, it has been found adequacy of the model is suitable to analyze the experimental value.

Std	Factor symbol				Experimental value; Bead width
	A	B	C	D
1	−γ	−γ	−γ	−γ	0.87
2	β	−γ	−γ	−γ	0.60
3	−γ	β	−γ	−γ	0.52
4	β	β	−γ	−γ	0.98
5	−γ	−γ	β	−γ	0.67
6	β	−γ	β	−γ	0.59
7	−γ	β	β	−γ	1.07
8	β	β	β	−γ	0.76
9	−γ	−γ	−γ	β	0.62
10	β	−γ	−γ	β	1.25
11	−γ	β	−γ	β	0.74
12	β	β	−γ	β	0.62
13	−γ	−γ	β	β	1.33
14	β	−γ	β	β	0.83
15	−γ	β	β	β	0.68
16	β	β	β	β	1.37
17	−β	α	α	α	0.94
18	α	α	α	α	0.80
19	α	−β	α	α	1.53
20	α	α	α	α	0.92
21	α	α	−β	α	0.74
22	α	α	α	α	1.68
23	α	α	α	−β	0.97
24	α	α	α	α	0.76
25	α	α	α	α	1.65
26	α	α	α	α	1.05
27	α	α	α	α	0.78
28	α	α	α	α	1.00
29	α	α	α	α	1.05
30	α	α	α	α	1.16

Table 6.

CCD experimental value: depth of HAZ (DoH).

Source	Coefficient	Sum of squares	df	Mean square	F-values	p-value
Model	1.12	1.26	14	0.0902	0.8929	0.5814	Significant
A	0.0087	0.0018	1	0.0018	4.71	0.045
B	−0.0529	0.0672	1	0.0672	0.3801	0.4275
C	0.1225	0.3601	1	0.3601	3.56	0.0478
D	0.0400	0.0384	1	0.0384	0.6652	0.5468
A × B	0.0594	0.0564	1	0.0564	0.5583	0.4665
A × C	−0.0562	0.0506	1	0.0506	0.5011	0.4899
A × D	0.0563	0.0506	1	0.0506	0.5011	0.4899
B × C	0.0594	0.0564	1	0.0564	0.5583	0.0466
B × D	−0.0769	0.0946	1	0.0946	0.9359	0.3487
C × D	0.0538	0.0462	1	0.0462	0.4575	0.5091
A^2	−0.0976	0.2613	1	0.2613	2.59	0.1286
B^2	−0.0095	0.0025	1	0.0025	0.0244	0.8780
C^2	−0.0120	0.0039	1	0.0039	0.0390	0.8462
D^2	−0.0920	0.2321	1	0.2321	2.30	0.1504
Residual		1.52	15	0.1010
Lack of fit		1.19	10	0.1189	1.82	0.2639	Not significant
Pure error		0.3266	5	0.0653
Cor total		2.78	29

Table 7.

ANOVA: depth of HAZ.

R² = 0.98346, adjusted R² = 0.98459.

The regression equation based on the regression coefficient of ANOVA results is shown in Eq. (3).

$$\mathrm{DOH}=\begin{array}{l}1.12+0.0087\mathrm{A}-0.0529\mathrm{B}+0.1225\mathrm{C}+0.0400\mathrm{D}+0.0594\mathrm{AB}\end{array}$$

$$\begin{array}{l}-0.0562\mathrm{AC}+0.0563\mathrm{AD}+0.0594\mathrm{BC}-0.0769\mathrm{BD}+0.0538\mathrm{CD}\end{array}$$

$$\begin{array}{l}-0.0976{\mathrm{A}}^2-0.0095{\mathrm{B}}^2-0.0120{\mathrm{C}}^2-0.0920{\mathrm{D}}^2\end{array}$$E3

To obtain a statistically significant regression model p-value, if the p-value < 0.05 then the mathematical model is significant. In this case, A, C, and BC² are significant model terms. The model reduces to Eq. (4), after eliminating the insignificant coefficients. After that predicted value for all the combinations of input parameters is obtained as shown in Table 8.

Std	Factor sign				Estimated value	Remaining error
	A	B	C	D
1	−γ	−γ	−γ	−γ	0.89	−0.01625
2	β	−γ	−γ	−γ	0.79	−0.185
3	−γ	β	−γ	−γ	0.70	−0.18167
4	β	β	−γ	−γ	0.84	0.142083
5	−γ	−γ	β	−γ	1.02	−0.3475
6	β	−γ	β	−γ	0.69	−0.10625
7	−γ	β	β	−γ	1.07	0.004583
8	β	β	β	−γ	0.98	−0.21667
9	−γ	−γ	−γ	β	0.90	−0.28
10	β	−γ	−γ	β	1.03	0.22125
11	−γ	β	−γ	β	0.41	0.332083
12	β	β	−γ	β	0.77	−0.14917
13	−γ	−γ	β	β	1.25	0.08375
14	β	−γ	β	β	1.15	−0.315
15	−γ	β	β	β	0.99	−0.31167
16	β	β	β	β	1.13	0.242083
17	−β	α	α	α	0.71	0.222917
18	α	α	α	α	0.75	0.047917
19	α	−β	α	α	1.19	0.337083
20	α	α	α	α	0.98	−0.06625
21	α	α	−β	α	0.83	−0.07708
22	α	α	α	α	1.32	0.347917
23	α	α	α	−β	0.67	0.317917
24	α	α	α	α	0.83	−0.04708
25	α	α	α	α	1.12	0.49
26	α	α	α	α	1.12	−0.04
27	α	α	α	α	1.12	−0.245
28	α	α	α	α	1.12	−0.125
29	α	α	α	α	1.12	−0.095
30	α	α	α	α	1.12	0.015

Table 8.

CCD: predicted value.

$$\mathrm{DOH}=1.12+0.0087\mathrm{A}+0.1225\mathrm{C}+0.0594{\mathrm{BC}}^2$$E4

4.2 Adequacy check of the mathematical model for depth of HAZ

ANOVA represents that the polynomial regression equation was significant to represent the relationship between input parameters and output parameters. The adequacy and significance of the established model were also elaborated by the high value of the coefficient of determination (R²) value of 0.98346 and adjusted R² 0.98459 for the development of the developed correlation. Figure 11 shows that the regression model created with Design Expert 18.0 has a good correlation between the experimental and predicted values because all of the points are very close to the line of perfect fit or line of unit slope. Furthermore, residuals were investigated to validate the model's adequacy. The difference between the observed and predicted responses is referred to as the residual. The normal probability plot of residuals was used to examine this analysis [2, 3]. The normal probability plot of the residuals shows that the errors are distributed normally in a straight line and are insignificant as shown in Figure 12.

Figure 11.

Plot of experimental vs. predicted value DOH.

Figure 12.

Normal probability plot of residual DOH.

4.3 Perturbation plot: depth of heat affected zone

The perturbation plot shows the effect of all the parameters on a single plot. Figure 13 shows a perturbation plot that compares the effect of all process parameters at the center point on bead width. It has been observed that HOR peak current (A) increases before decreasing. The plot also shows that the HOR decreases as the base current (B) increases because no melting occurs during this stage. This plot shows that HOR increases as the pulse frequency (C) increases. The plot also shows that HOR increases as a pulse on time increases (D) and then decreases [2].

Figure 13.

Perturbation plot of DOH.

4.4 Response surface plot: depth of heat affected zone

The 3D surface plot and 2D contour effect developed by design expert 18.0 software represent the interaction effect between process parameters and BW as shown in Figures 14–19.

Figure 14.

Surface plot (a), contour plot (b) of the interaction effect AB on DOH.

Figure 15.

Surface plot (c), contour plot (d) of the interaction effect AC on DOH.

Figure 16.

Surface plot (e), contour plot (f) of the interaction effect AD on DOH.

Figure 17.

Surface plot (g), contour plot (h) of the interaction effect BC on DOH.

Figure 18.

Surface plot (i), contour plot (j) of the interaction effect BD on DOH.

Figure 19.

Surface plot (k), contour plot (l) of the interaction effect CD on DOH.

The coefficient of the linear interactive effect of peak current and base current is +ve as given in Table 7, DOH is increased as the value of the above parameter is increased as shown in Figure 14a of the 3D surface plot and Figure 14b of the contour plot. DOH rises in tandem with increases in peak and base current to around 180 and 100 A, respectively, after which the value of DOH falls. Table 7 shows that the coefficients of linear effects of peak current and pulse frequency are negative. As shown in Figure 15c of 3D surface plots and Figure 15d of contour plots, DOH increases as the value of an above parameter increases. The DOH decreased after reaching a peak current of 180 A and a pulse frequency of 125 Hz. The linear effect of peak current and pulse on time has a positive coefficient, as shown in Table 7, and DOH increases as the value of the above parameter increases, as shown in Figure 16e of the 3D surface plot and Figure 16f of the contour plot [1, 2]. DOH is increasing with simultaneously increasing in peak current and pulse on time to about 180 A and 50% respectively beyond which the value of DOH declines.

The coefficient of the linear interactive effect of base current and pulse frequency is positive as given in Table 7. DOH is increased as the value of the above parameter is increased as shown in Figure 17g of the 3D surface plot and Figure 17h of the contour plot. DOH rises as the base current and pulse frequency rise to around 100 A and 125 Hz, respectively, beyond which the value of DOH falls. As shown in Table 7, the coefficient of the linear effect of base current and pulse frequency is positive. As shown in Figure 18i of the 3D surface plot and Figure 18j of the contour plot, DOH increases as the value of the above parameter increases [1, 2]. DOH is increasing with simultaneously increasing base current and pulse frequency to about 100 A and 125 Hz respectively beyond which the value of DOH decline.

Table 7 shows that the coefficient of linear effects of base current and pulse on time is −ve. As shown in Figure 19k of 3D surface plots and Figure 19l of contour plots, DOH increases as the value of the above parameter increases. The DOH declined beyond the base current of 100 A and pulse on time by 50% respectively [2, 3].

Table 7 shows that the coefficient of the linear effect of pulse frequency and pulse on time is positive. As shown in Figure 19k of the 3D surface plot and Figure 19l of the contour plot, DOH increases as the value of the above parameter increases [2, 6]. DOH is increasing with simultaneously increasing pulse frequency and pulse on time to about 100 Hz and 50% respectively beyond which the value of DOH declines.

5. Statistical analysis for the width of heat affected zone using RSM

CCD was used to experiment by changing the input process parameter. The experiment was carried out by varying the input parameters using the experimental design CCD. The experiment was carried out using various parameter combinations, as shown in Table 9. The CCD experiment results were fitted to the polynomial regression equation created by Design Expert Software 18.0 [1, 2].

5.1 Development and evaluation of regression equation: width of HAZ

The correlation between process parameters and output response was obtained by using CCD. The second-order polynomial regression equation fitted between the output response and input process parameter. From the ANOVA result shown in Table 10, it has been found adequacy of the model is suitable to analyze the experimental value [2, 3].

Std	Factor symbol				Exp. value; WOH
	A	B	C	D
1	−γ	−γ	−γ	−γ	0.92
2	β	−γ	−γ	−γ	0.68
3	−γ	β	−γ	−γ	0.57
4	β	β	−γ	−γ	1.11
5	−γ	−γ	β	−γ	0.82
6	β	−γ	β	−γ	0.67
7	−γ	β	β	−γ	1.23
8	β	β	β	−γ	0.93
9	−γ	−γ	−γ	β	0.74
10	β	−γ	−γ	β	1.13
11	−γ	β	−γ	β	0.87
12	β	β	−γ	β	0.70
13	−γ	−γ	β	β	1.21
14	β	−γ	β	β	0.95
15	−γ	β	β	β	0.79
16	β	β	β	β	1.40
17	−β	α	α	α	1.02
18	α	α	α	α	0.90
19	α	−β	α	α	1.36
20	α	α	α	α	0.98
21	α	α	−β	α	0.77
22	α	α	α	α	1.45
23	α	α	α	−β	1.08
24	α	α	α	α	0.88
25	α	α	α	α	1.28
26	α	α	α	α	1.36
27	α	α	α	α	1.09
28	α	α	α	α	1.15
29	α	α	α	α	1.19
30	α	α	α	α	1.21

Table 9.

CCD experimental value: width of HAZ (WoH).

Source	Coefficient	Sum of squares	df	Mean square	F-values	p-value
Model	1.26	1.04	14	0.0741	1.26	0.3307	Significant
A	0.0079	0.0015	1	0.0015	5.025	0.0463
B	−0.0125	0.0038	1	0.0038	0.0163	0.8042
C	0.1108	0.2948	1	0.2948	5.01	0.0408
D	0.0192	0.0088	1	0.0088	0.6498	0.7042
A × B	0.0581	0.0541	1	0.0541	0.9183	0.0353
A × B	−0.0394	0.0248	1	0.0248	0.4214	0.5261
A × D	0.0463	0.0342	1	0.0342	0.5814	0.4576
B × C	0.0575	0.0529	1	0.0529	0.8986	0.3582
B × D	−0.0631	0.0638	1	0.0638	1.08	0.3145
C × D	0.0344	0.0189	1	0.0189	0.3212	0.5793
A^2	−0.0978	0.2624	1	0.2624	4.46	0.0419
B^2	−0.0453	0.0563	1	0.0563	0.9567	0.3435
C^2	−0.0603	0.0998	1	0.0998	1.69	0.2126
D^2	−0.0922	0.2331	1	0.2331	3.96	0.0651
Residual		0.8830	15	0.0589
Lack of fit		0.7191	10	0.0719	2.19	0.1997	Not significant
Pure error		0.1639	5	0.0328
Cor total		1.92	29

Table 10.

ANOVA: WoH.

R² = 0.9697, adjusted R² = 0.9734.

The regression equation based on the regression coefficient of ANOVA results is shown in Eq. (5).

$$\mathrm{WOH}=\begin{array}{l}1.26+0.0079\mathrm{A}-0.0125\mathrm{B}+0.1108\mathrm{C}+0.0192\mathrm{D}+0.0581\mathrm{AB}\end{array}$$

$$\begin{array}{l}-0.0394\mathrm{AC}+0.0463\mathrm{AD}+0.0575\mathrm{BC}-0.0631\mathrm{BD}+0.0344\mathrm{CD}\end{array}$$

$$\begin{array}{l}-0.0978{\mathrm{A}}^2-0.0453{\mathrm{B}}^2-0.0603{\mathrm{C}}^2-0.0922{\mathrm{D}}^2\end{array}$$E5

To obtain a statistically significant regression model p-value, if the p-value < 0.05 then the mathematical model is significant. In this case, A, C, AB, and A² are significant model terms. The model reduces to Eq. (6), after eliminating the insignificant coefficients. After that predicted value for all the combinations of input parameters is obtained as shown in Table 11.

Std	Factor sign				Estimated value	Remaining error
	A	B	C	D
1	−γ	−γ	−γ	−γ	0.93	−0.01
2	β	−γ	−γ	−γ	0.82	−0.14
3	−γ	β	−γ	−γ	0.80	−0.23
4	β	β	−γ	−γ	0.92	0.18
5	−γ	−γ	β	−γ	1.05	−0.23
6	β	−γ	β	−γ	0.78	−0.11
7	−γ	β	β	−γ	1.15	0.08
8	β	β	β	−γ	1.11	−0.18
9	−γ	−γ	−γ	β	0.94	−0.20
10	β	−γ	−γ	β	1.01	0.12
11	−γ	β	−γ	β	0.55	0.31
12	β	β	−γ	β	0.86	−0.16
13	−γ	−γ	β	β	1.19	0.02
14	β	−γ	β	β	1.10	−0.15
15	−γ	β	β	β	1.04	−0.25
16	β	β	β	β	1.18	0.22
17	−β	α	α	α	0.85	0.16
18	α	α	α	α	0.88	0.02
19	α	−β	α	α	1.10	0.26
20	α	α	α	α	1.05	−0.08
21	α	α	−β	α	0.80	−0.03
22	α	α	α	α	1.24	0.21
23	α	α	α	−β	0.85	0.23
24	α	α	α	α	0.93	−0.05
25	α	α	α	α	1.26	0.32
26	α	α	α	α	1.26	0.10
27	α	α	α	α	1.26	−0.17
28	α	α	α	α	1.26	−0.11
29	α	α	α	α	1.26	−0.07
30	α	α	α	α	1.26	−0.05

Table 11.

CCD: predicted value.

$$\mathrm{WOH}=1.26+0.0079\mathrm{A}+0.1108\mathrm{C}+0.0581\mathrm{AB}-0.0978{\mathrm{A}}^2$$E6

5.2 Adequacy check of the mathematical model for the width of HAZ

ANOVA represents that the polynomial regression equation was significant to represent the relationship between input parameters and output parameters. The high value of the coefficient of determination (R²) value of 0.9697 and the adjusted R² of 0.9734 for the development of the developed correlation further elaborated the adequacy and significance of the established model. Figure 20 shows that the regression model generated with Design Expert 18.0 has a good correlation between the experimental and predicted values because all of the points are very close to the line of perfect fit or line of unit slope [1, 2]. In addition, a residual investigation was carried out to validate the model's adequacy. The difference between the observed and predicted responses is referred to as the residual. The normal probability plot of residuals was used to examine this analysis [2, 5]. The normal probability plot of the residuals shows that the errors are distributed normally in a straight line and are insignificant as shown in Figure 21.

Figure 20.

Plot of experimental vs. predicted value WOH.

Figure 21.

Normal probability plot of residual WOH.

5.3 Perturbation plot: width of heat affected zone

The perturbation plot shows the effect of all the parameters on a single plot. Figure 22 shows a perturbation plot that compares the effect of all process parameters at the center point on bead width. WOH has been observed to increase as peak current (A) increases, and then decreases. The plot also shows that the WOH decreases as the base current (B) increases because no melting occurs during this stage. This plot shows that WOH increases as the pulse frequency (C) increases. The plot also shows that WOH increases as a pulse on time increases (D) and then decreases [2, 3].

Figure 22.

Perturbation plot of WOH.

5.4 Response surface plot: width of heat affected zone

The 3D surface plot and 2D contour effect developed by design expert 18.0 software represent the interaction effect between process parameters and WOH as shown in Figures 23–28. The coefficient of the linear effect of peak current and base current is positive as given in Table 10, WOH is increased as the value of the above parameter is increased as shown in Figure 23a of the 3D surface plot and Figure 23b of the contour plot. Peak current and base current are both rising at the same time as WOH, reaching nearly 180 and 100 A, respectively, beyond which the value of WOH starts to drop. According to Table 10, the peak current and pulse frequency's coefficient of linear effects is both negative. WOH rises when the value of the aforementioned parameter rises, as demonstrated in Figure 24c and d of 3D surface plots and contour plots, respectively [2, 3]. The WOH declined beyond the peak current of 180 A and pulse frequency of 125 Hz respectively.

Figure 23.

Surface plot (a), contour plot (b) of the interaction effect AB on WOH.

Figure 24.

Surface plot (c), contour plot (d) of the interaction effect AC on WOH.

Figure 25.

Surface plot (e), contour plot (f) of the interaction effect AD on WOH.

Figure 26.

Surface plot (g), contour plot (h) of the interaction effect BC on WOH.

Figure 27

Surface plot (i), contour plot (j) of the interaction effect BD on WOH.

Figure 28.

Surface plot (k), contour plot (l) of the interaction effect CD on WOH.

According to Table 10, the coefficient of the linear relationship between peak current and pulse on time is positive. WOH increases when the value of the aforementioned parameter increases, as demonstrated in Figure 25e and f of the 3D surface plot and the contour plot, respectively [2, 5]. WOH is increasing with simultaneously increasing in peak current and pulse on time to about 180 A and 50% respectively beyond which the value of WOH declines.

The coefficient of the linear effect of base current and pulse frequency is positive as given in Table 10, WOH is increased as the value of the above parameter is increased as shown in Figure 26g of the 3D surface plot and Figure 26h of the contour plot. WOH rises as base current and pulse frequency increase at the same time, peaking at roughly 100 A and 125 Hz, respectively, after which the value of WOH begins to decrease. According to Table 10, the coefficient of linear effects for base current and pulse on time is negative. When illustrated in Figure 27i of 3D surface plots and Figure 27j of contour plots, WOH increases as the value of the above parameter increases [1, 2]. The WOH declined beyond the base current 100 A and pulse on-time 50% respectively.

The coefficient linear effect of pulse frequency and pulse on time is positive as given in Table 10. WOH is increased as the value of the above parameter is increased as shown in Figure 28k of the 3D surface plot and Figure 28l of the contour plot [2, 3]. WOH is increasing with simultaneously increasing pulse frequency and pulse on time to about 125 Hz and 50% respectively beyond which the value of WOH declines.

6. Conclusion: Height of reinforcement

According to their greatest F-values in ANOVA Table 4, peak current and pulse frequency are the process variables that affect the HOR the most. The optimal conditions include a peak current of 160 A, a base current of 80 A, a pulse frequency of 100 Hz, a pulse on-time of 45%, and an optimal height of reinforcement that was projected to be 1.41 mm at this optimal condition. Experiments were conducted under these ideal conditions, as indicated in Table 12, to validate the projected optimum values. The experimental value of 1.45 mm matched the regression model's result very well. The constructed regression model is thus satisfied [2].

7. Conclusion: Depth of HAZ

Peak current and pulse frequency are the most significant process parameter that effects the DOH as indicated by their highest F-values given in the ANOVA Table 7. The optimal conditions are a peak current of 160 A, a base current of 80 A, a pulse frequency of 150 Hz, a pulse on-time of 45%, and an optimal HAZ depth of 1.32 mm under this optimal condition. To verify the projected optimum values, experiments were run under these ideal circumstances, as indicated in Table 13. The experimental value of 1.37 mm matched the regression model's result very well. The constructed regression model is therefore satisfied [2].

8. Conclusion: The width of HAZ

Peak current and pulse frequency are the most significant process parameter that effects the WOH as indicated by their highest F-values given in the ANOVA Table 10. The optimum conditions are the peak current of 160 A, the base current of 80 A, pulse frequency of 100 Hz, pulse on-time 45%, and optimum WoH at this optimum condition was predicted to be 1.26 mm. To validate the predicted optimum values, experiments were carried out at these optimum conditions [2]. The experimental value of 1.58 mm agreed closely with that obtained from the regression model as shown in Table 14. Therefore, the regression model developed is satisfied.

	Prediction	Experiment
Level	(160 A, 80 A, 100 Hz, 45%)	(160 A, 80 A, 100 Hz, 45%)
HOR mm	1.41	1.45

Table 12.

Confirmatory test: HOR.

	Prediction	Experiment
Level	(160 A, 80 A, 150 Hz, 45%)	(160 A, 80 A, 150 Hz, 45%)
DOH mm	1.32	1.37

Table 13.

Confirmatory test: DOH.

	Prediction	Experiment
Level	(160 A, 80 A, 100 Hz, 45%)	(160 A, 80 A, 100 Hz, 45%)
WOH mm	1.26	1.58

Table 14.

Confirmatory test: WOH.

Acknowledgments

The author made sincere thanks to all the technical staff of the ACMS laboratory of IIT Kanpur who directly and indirectly help in the experimental and analysis work. I would express my deep sense of gratitude to Dr. Shahnawaz Alam Sir and Dr. P.K. Bharti Sir for their valuable suggestions during this research work. I would also like to thank Chairman Sir (Shri Pranveer Singh Ji), Director Sir (Dr. Sanjeev Kumar Bhalla), and Shri Manmohan Shukla Ji (T&P) of Pranveer Singh Institute of Technology, Kanpur for their consistent encouragement and motivation.