Effect of operating parameters on plate bending by laser line heating

Abstract

Laser line heating is a plate bending technique. It is produced due to the heating effect of a laser beam when irradiated over a suitable heating path. In this work, the effect on angular deformation under different operating parameters, such as energy (in terms of laser power), scanning speed and number of passes along with the thickness of the substrate material, was studied under straight line scanning schemes. In this experiment, CO₂ laser has been used and the substrate material used is the mild steel sheet. The effect of each operating parameter was taken into account to find the optimum process parameter. The design of experiment has been applied with the help of Taguchi method, and the results were obtained and analysed with the help of analysis of variance and Taguchi analysis (signal-to-noise ratio), for the determination of the optimized values of the operating parameters with their effect towards angular deformation in the process. A regression analysis also has also been performed to obtain a suitable co-relationship between bending angle and other operating parameters.

Introduction

The process of line heating assisted with laser is found to be one of the most versatile techniques for various surface development activities. The process is based on localized heating without increasing the temperature beyond recrystallization point. The mechanism of the process is dependent on various process parameters, the shape of the substrate and the material properties to be used for the substrate. Here, in this case the temperature gradient mechanism (TGM) has been incorporated. In this case, bending is induced by thermal gradient between surfaces of the material to be processed. Temperature gradient is the reason for bending of thick plates; various analytical methods are proposed for the determination of temperature field which is required for the simulation of the line heating technique.

Zhang et al. ¹ presented the effect of line heating factors including heat input, heating line position from plate edge and plate size on angular deformation which were explored with the numerical simulation. The main motto is to develop a relation between the heating parameters and the angular deformation, which are required for fabrication of ship hull by the process of line heating.

For the occurrence of TGM, the value of the Fourier number (F₀) has to be lesser than unity. ² The Fourier number (F₀) can be expressed as follows

$$F_0=\frac{\psi d}{t^{2}v}$$

where Ψ, d, t and v are thermal diffusivity, laser beam diameter, sheet thickness and scanning velocity, respectively).

According to Lawrence et al., ³ for TGM to occur, laser spot diameter is somehow equal to that of the thickness of the sheet. The effect of bending angle with reference to the change in laser parameters has been proposed by Shichun and Jinsong. ⁴ They proposed that the increase in angular deformation occurs with the rise in laser power, number of scans, thickness of the material and thermal effect index and decrease with higher spot diameter, thickness of the sheet and scanning speed. The analysis based on TGM was proposed by Shen et al. ⁵ by taking the cross section into consideration. An equation was proposed for the prediction of angular deformation with the variation in laser power and scan speed.

Vollertsen ⁶ has developed a relation for the determination of angular deformation. The expression was obtained on the basis of TGM. The expression for the bending angle was obtained on the basis of process parameters and material properties. Similar formulation was also given by Yau et al., ⁷ where the expression of bend angle is obtained based on two-layer model for TGM along with addition of counter bending effect for some of the pure elastic strains, which resulted into dual equations, for obtaining both counter bending and bending angle. Mucha et al. ⁸ have derived an expression for obtaining the bending angle; the expression obtained is based on TGM. The output obtained reveals that the coefficient of thermal expansion and heat capacity has a significant effect as material parameters with reference to that of laser forming process. Kyrsanidi et al. ⁹ have developed a parametric mathematical model where plastic bending is considered at the time of heating. It is also found that the developed model took very less computing time in comparison to the existing model.

Shen et al. ¹⁰ have derived a formula for obtaining angular deformation by laser forming process. A model was developed on the basis of conventional equilibrium and compatibility conditions. Mucha ¹¹ has developed a model for the process of plate bending on the basis of TGM. The model provides solution for longitudinal and transversal angular deformations. Shi et al. ¹² present the study of TGM for the forming process. Under processing condition of TGM, the plate bends in the direction of both x- and y-axes. The model has been developed for estimating the angular deformation about y-axis, which has been obtained based on the heat transfer theories and mechanics of elasto-plasticity. Shen et al. ¹³ also proposed a work related to the forming of plates under the action of laser as a heat source under simultaneous scanning of laser in parallel directions. The outcome reveals that the angular deformations were obtained for both single and multiple scanning under the same line, keeping less gap between the two scanning paths. Casalino and Ludovico ¹⁴ have proposed a feed forward neural network with back propagation technique for evaluating bending angle and for selecting the TGM and buckling mechanism (BM) laser forming parameters. The result reveals that neural network provides fast and precise outcome in comparison to finite element analysis and it is easier to use them than the multivariate regression analysis.

In this article, an experimental study was conducted for obtaining the optimized values of operating parameters which include scanning speed, laser power and the number of scans on the angular deformation with the change in thickness of the substrate material based on TGM. The output results obtained were suitably analysed under Taguchi analysis and analysis of variance (ANOVA), and with the development of relationship between the operating parameters, suitable optimum parameters were obtained.

Experimental details

A CO₂ gas laser machine (Orion 3015, LVD make) was used for the experiments. Rectangular mild steel blanks of dimension of 200 mm × 200 mm and thicknesses of 3 and 4 mm were taken as the working specimens. For the experiment to be started, the specimens were put on the machine work table with the help of clamp, with job holding attachment as shown in Figure 1; about 5–6 mm gap was maintained between the work table and the specimen so that there should be no constraints for the specimen while undergoing counter bending. Numerous numbers of experiments were carried out by altering the operating parameters. The angular deformation of the plate was determined with the help of coordinate measuring machine (CMM) for individual set of experiments.

Figure 1.

Experimental set-up for straight line bending.

For calculating the stand-off distance, initially the width of the zone (w) affected by moving heat source is taken into account along with the uniform velocity (v). It is estimated to be equal to the summation of the diameter of the laser beam (d) and the thermal diffusion length ( $l^{*}$ ). The thermal diffusion length ( $l^{*}$ ) can be calculated as

$$l^{*}=2\sqrt{\frac{\psi \times d}{v}}$$

(1)

where ‘ $\psi$ ’ is the thermal diffusivity, which can be again expressed as

$$\psi =\frac{k}{\rho \times C}$$

(2)

where ‘k’ is the thermal conductivity of the material, ‘ $\rho$ ’ is the density of the material and ‘C’ is the specific heat capacity of the material.

Therefore, the width of the zone (w) can be expressed as

$$w=d+2\left(2\sqrt{\frac{\psi \times d}{v}}\right)=d+4\left(\sqrt{\frac{\psi \times d}{v}}\right)$$

(3)

From the above expression, if the velocity (v) is taken to be very large, then the width of the zone (w) can be considered to be the measure of beam diameter (d). On the basis of this theory, it is assumed that the heating impression is obtained with the help of moving laser beam at high speed (probably be the highest speed possible, speed attainable with the work station they worked with) to be the beam diameter (d) at a particular stand-off distance between the tip of the nozzle and the workpiece surface. As in this case we need to maintain the TGM, the laser beam diameter is to be equal to that of the thickness of the substrate (workpiece). For this reason, suitable stand-off distance was maintained. The rectangular plates were heated along the centre line of the top surface of the plates.

Taguchi method of design of experiment

The design of experiment (DOE) is a method where the experiments were performed according to a fixed system or plan. This is necessary for solving any problem related to engineering in principles, techniques and data collection stage, ensuring in getting some logical and justifiable conclusions. It also enables the designs for the determination of individual and interactive effects of many factors, which could affect the output results in any design. Here in this case, Taguchi method of DOE is taken into consideration before performing the experiments and analysis.

Taguchi method ¹⁵ has been designed for obtaining improved quality processes and outputs where the performance of the output depends on several factors. It can be thought that for carrying out any test and development, simple logic is sufficient for establishment of possible combination of several factors along with the ranges to be taken into consideration. Taguchi has developed a set of designs for applications. The special features of this type of designs are the usage of orthogonal arrays (OAs). These arrays help in performing minimum number of experiments required for a given set of parameters. Here, we have used Taguchi DOE for optimizing the operating parameters. The experiment design must satisfy the following two conditions:

1. The number of trial runs must be determined.

2. Trial conditions are to be specified.

Selection of the level of parameters

For the selection of the level of parameters, the initial selection of the number of operating parameters involves in the process and then the level is to be defined. If it happens that the number of levels is not uniform for all, then the mixed level design is to be taken into consideration. By considering both the values (number of parameters and levels), the suitable OA has to be chosen from the manuals or handbook. The design should be extracted from the listed designs and it is to be used.

Here, the experimental design for multi-pass line heating was taken into consideration by selecting four laser operating parameters; they are as follows: traverse speed, laser power, plate thickness and number of scans. All the parameters were varied in four levels except the plate thickness which is varied in two levels. L-16 OA was incorporated for the DOE as shown in Table 1, and with the help of parameter and output signal-to-noise (S/N) ratio and S/N analysis were performed. As in this case the main aim is obtaining large bending with the change in operating parameters, higher values of S/N ratio give better result corresponding to optimal process parameters.

Table 1.

Operating parameters and their selected levels (under Taguchi DOE L-16 array).

Experiment No.	Traverse speed (mm/min)	Power (W)	Number of passes	Sheet thickness (mm)
1	500	300	1	3
2	500	400	2	3
3	500	500	3	4
4	500	600	4	4
5	400	300	2	4
6	400	400	1	4
7	400	500	4	3
8	400	600	3	3
9	300	300	3	3
10	300	400	4	3
11	300	500	1	4
12	300	600	2	4
13	200	300	4	4
14	200	400	3	4
15	200	500	2	3
16	200	600	1	3

Results and discussion

Single-pass line heating

A series of experiments on single pass (run) were performed for obtaining the angular deformation of mild steel plates of dimension 200 mm × 200 mm × 3 mm. The experiments were carried out by altering the laser power and traverse speed, keeping the laser diameter constant, that is, 3 mm. The peak temperatures obtained from the experimental study at different locations (i.e. as shown in Figure 2) are shown in Table 2 which shows the effect of operating parameters on peak temperatures at different locations for single-pass line heating process. The results are shown in Table 3 which shows the effect of angular deformation for single-pass line heating process.

Figure 2.

Positions of thermocouples.

Table 2.

Peak temperature in thermocouple locations under single-pass line heating.

Exp No.	Power (W)	Traverse speed (mm/min)	Temperature at position ‘A’ (°C)	Temperature at position ‘B’ (°C)	Temperature at position ‘C’ (°C)	Temperature at position ‘D’ (°C)
1	400	500	142.027	97.266	99.665	146.963
2	400	400	164.923	115.136	121.176	168.714
3	400	300	207.066	154.467	157.59	215.56
4	400	200	262.974	201.42	223.14	300.519
5	500	500	220.825	49.419	139.485	224.79
6	500	400	226.86	59.374	182.369	343.546
7	600	500	147.455	103.694	146.77	380.806
8	600	200	372.536	238.252	309.375	523.17

Table 3.

Results for experiments under single-pass line heating.

Experiment No.	Power (W)	Traverse speed (mm/min)	Angular deformation for single pass (°)
1	400	500	0.2502
2	400	400	0.8084
3	400	300	0.9666
4	400	200	1.026
5	500	500	0.81328
6	500	400	1.1901
7	500	300	1.2799
8	500	200	1.3999
9	600	500	1.1133
10	600	400	1.3667
11	600	300	1.1496
12	600	200	1.652

Figure 3 shows the curve between the bend angles versus traverse speeds for mild steel plate of 3 mm thickness. The values obtained were under different sets of power and traverse speed. It has been observed that constant power and increase in traverse speed lead to the reduction in angular deformation (bend angle). Another study has been performed based on the trend line which has been obtained based on the values of bend angle obtained under different heat inputs per unit length, as shown in Figure 4, for single-pass line heating process. Here, the heat inputs per unit length have been calculated using laser power (P) and traverse speed (v), that is, P/v.

Figure 3.

Traverse speed versus bending angle for different laser powers.

Figure 4.

Heat input per unit length versus bending angle.

From Figure 3, it is also revealed that with uniform increment of laser power under constant traverse speed, the angular deformation is not uniform. With the increase in power, the behaviour follows non-uniformly. From Figure 4, it has been observed that with the increase in line heat input, that is, heat input per unit length, the angular deformation increases. But the rate of increase in angular deformation is more with lower values of line heat input. After a certain line heat input (i.e. approximately 150 J/mm), the rate of increase in angular deformation is very low. So from Figure 4, it can be concluded that the line heating should be carried out near the line heat input value of 150 J/mm. So in this way for different thicknesses of plates, the optimum line heat input can be calculated.

Multi-pass line heating

We have developed the design of experiment on the Taguchi DOE, L-16 array, on the basis of that experimentation has been done and the corresponding results of bending angle are tabulated in Table 4. Figure 5 represents the heat input per unit length versus bend angle under different scanning schemes for mild steel plate of 3 and 4 mm thicknesses.

Table 4.

Experimental results on the basis of Taguchi L-16 array DOE.

Experiment No.	Traverse speed (mm/min)	Power (W)	Number of passes	Plate thickness (mm)	Resultant angular deformation (°)
1	500	300	1	3	0.65
2	500	400	2	3	1.49010
3	500	500	3	4	1.42505
4	500	600	4	4	3.00825
5	400	300	2	4	0.04000
6	400	400	1	4	0.17920
7	400	500	4	3	4.59330
8	400	600	3	3	3.61996
9	300	300	3	3	2.69666
10	300	400	4	3	4.24840
11	300	500	1	4	0.64720
12	300	600	2	4	1.30280
13	200	300	4	4	2.52780
14	200	400	3	4	2.54715
15	200	500	2	3	2.58338
16	200	600	1	3	0.77000

Figure 5.

Heat input per unit length versus bend angle under different scanning schemes for 3 and 4 mm thicknesses of plates.

With the increase in material thickness and strength, more amount of energy is required for obtaining the same angular deformation. Here in this technique as multiple passes are involved, it has been observed that with the increase in the number of passes, the angular deformation increases. This is because the retained heat input in the irradiated area from the first pass could serve for the additional forming by reducing the temperature-dependent flow stress of the material, in that a hot plate is easier to form than a cold one. Figure 5 shows the results obtained under multiple scanning schemes under different heat inputs per unit length for samples of 3- and 4-mm-thick mild steel plates.

Analysis of DOE results

The obtained experimental results were investigated under Taguchi method of analysis. Taguchi analysis incorporates the basic three objectives, which are as follows: (1) for the determination of the optimum condition, (2) factors contributing to the results and determination of percentage contribution of the operating parameters to the output results and (3) the expected results at the optimum condition.

Identification of the optimum condition can be obtained by simple arithmetic calculations. ¹⁵ On determination of optimum condition, trial run must be accompanied so as to know the difference in the output in the experiment conducted in optimum and non-optimum condition. Optimum may not be the set of experiment that has been obtained under OA.

S/N and its significance

The ‘signal’ depicts the useful information and the ‘noise’ represents the value which is undesirable. ¹⁶ The expression for the S/N ratio has been formulated in such a way that the higher factor level settings are being utilized for obtaining the optimized quality characteristics of the experiment.

From the Taguchi analysis, we obtain the plot for S/N ratio in Figure 6. The values of the graph indicate the optimal parameters based on the bending angle as an output for the process.

Figure 6.

Mean S/N ratio for the input parameters.

From Figure 6, it has been observed that the usage of the values of the parameter having negative mean S/N ratio value will lead in obtaining poor quality of output (bend angle). For obtaining the optimum bend angle, the parameters should be selected in the positive region of vertical axis for all parameters. It is observed from Figure 5 that optimum bend angle can be obtained using laser power = 600 W, traverse speed = 200 mm/min, number of passes = 4 and sheet thickness = 3 mm.

ANOVA

It is a technique which has been used for providing measure of confidence. This technique has been applied to the experimental results for the determination of the effectiveness of the parameters that are responsible for the required output. This technique helps in providing confidence on the basis of variance of the data. ¹⁶

From the above analysis as shown in Table 5, the R² and the adjustable R² are found to be about 95.62% and 86.85%, respectively, which indicate the wellness of data to fit into a statistical model.

Table 5.

ANOVA table for bending angle.

Source	DF	Seq. SS	Adj. SS	Adj. MS	F	p	Percentage contribution
Traverse speed	3	0.7946	0.7946	0.2649	1	0.465	2.629723889
Laser power	3	1.647	1.647	0.549	2.07	0.223	5.450736528
Number of passes	3	21.4155	21.4155	7.1385	26.94	0.002	70.87446759
Sheet thickness	1	5.0341	5.0341	5.0341	19	0.007	16.66032347
Residual error	5	1.3248	1.3248	0.265	–	–	4.384417579
Total	15	30.2161	–	–	–	–	–

DF: degree of freedom; Seq. SS: sequential sum of square; Adj. SS: adjusted sum of square; Adj. MS: adjusted mean of square.

From Figure 7, it can be observed that the bending angle has a significant influence on sheet thickness and number of passes (scans) than the other parameters. It is also observed from the values of percentage contribution that both scanning speed and the number of passes are dominant with respect to all other parameters.

Figure 7.

Percentage contribution of individual operating parameters.

Regression analysis

A multivariable regression analysis has been done for obtaining a relationship between the angular deformation and the input parameters (i.e. laser power, traverse speed, sheet thickness and number of passes). The equation obtained based on the regression analysis is shown in equation (4)

$$\varnothing =2.87-0.00151v+0.00229P+1.03N-1.12t$$

(4)

where $\varnothing$ is the bend angle (°), $v$ is the traverse speed (mm/min), P is the laser power (W), N is the number of passes and t is the sheet thickness (mm).

Here, R² acts as a measure of accuracy of the result to be obtained as output in relation to that of the input parameters with the experimental results. Higher value of R² indicates that the empirical relation obtained, that is, equation (4), above has a better predicting capability of the output (bend angle), and there will be less mismatch between the experimental output and the output obtained from equation (4). As per Table 6, which is the regression table for equation (4), the value of R² is found to be 92.1%, which shows that the model can be used with sufficient accuracy.

Table 6.

Regression table.

Predictor	Coef.	SE coef.	T	p
Constant	2.867	1.05	2.73	0.02
Traverse speed	−0.00151	0.001044	−1.44	0.177
Laser power	0.002286	0.001044	2.19	0.051
Number of passes	1.0316	0.1044	9.88	0
Sheet thickness	−1.1218	0.2334	−4.81	0.001
S = 0.466887, R2 = 92.1%, R2(adj) = 89.2%

SE: standard error.

A test run has been performed based on the parameter tabulated in Table 7 and it was observed that the experimental output obtained validates well with that of the output obtained from the regression equation (i.e. equation (4)). The results are also plotted in Figure 8. The results obtained are of great accuracy with minimal amount of percentage of error. Thus, this model can be used for predicting the bend angle by altering the input parameters within the range.

Table 7.

Values of the bend angle obtained from the regression equation.

Experiment No.	Traverse speed (mm/min)	Laser power (W)	No. of pass	Plate thickness (mm)	Bend angle from experiment	Bend angle from regression analysis equation	Percentage of error
1	400	400	1	3	0.80	0.852	6.103
2	300	400	1	3	0.96	1.003	4.287
3	400	700	1	4	0.45	0.419	7.398
4	500	600	1	4	0.037	0.039	5.128
5	600	450	2	4	0.523	0.5746	8.859
6	200	600	4	3	5.10	4.702	8.464

Figure 8.

Comparative plot between the bending angles obtained from experiment and regression equation.

Conclusion

In this article, bending of mild steel sheet is carried out with the process of laser line heating with the input process parameters considered as traverse speed, laser power, number of passes and sheet thickness, and the response obtained is bend angle as the output.

• The line heating process parameters are optimized with respect to multiple performances in order to achieve best quality of angle of bend.

• Optimization of the process parameters was carried out using Taguchi method of analysis and a statistical technique called ANOVA.

• It was identified that laser power of 600 W, traverse speed of 200 mm/min and number of passes of four for sheet thickness of 3 mm are the optimal combinations of laser line heating process parameters that produce an output of 5.1° as an angle of bend.

• ANOVA statistics revealed that the number of passes is the most influencing parameter in achieving good results followed by sheet thickness, laser power and traverse speed.

• A suitable co-relationship has been obtained between the laser power, traverse speed, sheet thickness and number of passes with bend angle using multiple regression analysis. The expression validates well with the experimental results.

• Thus, it is concluded that the procedure proposed in this article for optimization of laser line heating process parameters will significantly improve the angle of bend for mild steel sheets and plates.