The main qualities after laser drilling on PMMA plate are confirmed from specified measurement data. The shape of the drilled hole might be irregular. To facilitate the definition for quality, the characteristics of the drill holes are classified into the following categories. The measurement methods and quality calculation will be described in this section.
3 − 1 The Roundness of the drilled hole
To evaluate the roundness of a laser drilled hole, the irregular hole is considered as elliptical shape, as shown in Table 2. Both elliptical shapes at entrance side (Een) and exit side (Eex) are taken into account. The Roundness calculation for Entrance Side is obtained with Eq. (1).
Table 2
Experimental design using L9 (3)4 and four control factors
| Run No. | A | B | C | D | Laser energy (mJ/pulse) | Focusing position offset (mm) | Drill time (min) | Repetition rate(kHz) |
| 1 | 1 | 1 | 1 | 1 | 75 | + 1 | 3 | 1.8 |
| 2 | 1 | 2 | 2 | 2 | 75 | 0 | 5 | 3.0 |
| 3 | 1 | 3 | 3 | 3 | 75 | -1 | 7 | 4.2 |
| 4 | 2 | 1 | 2 | 3 | 110 | + 1 | 5 | 4.2 |
| 5 | 2 | 2 | 3 | 1 | 110 | 0 | 7 | 1.8 |
| 6 | 2 | 3 | 1 | 2 | 110 | -1 | 3 | 3.0 |
| 7 | 3 | 1 | 3 | 2 | 145 | + 1 | 7 | 3.0 |
| 8 | 3 | 2 | 1 | 3 | 145 | 0 | 3 | 4.2 |
| 9 | 3 | 3 | 2 | 1 | 145 | -1 | 5 | 1.8 |
$${{E}}_{{e}{n}}^{}=\frac{{{b}}_{{e}1}^{}}{{{a}}_{{e}1}^{}}$$
1
Where be1 is the shorter or minor elliptical diameter at entrance side (mm). ae1 is the longer or major elliptical diameter at entrance side (mm).
Similarly, the calculation for roundness of drilled hole at exit side is obtained with Eq. (2).
$${{E}}_{{e}{x}}^{}=\frac{{{b}}_{{x}2}^{}}{{{a}}_{{x}2}^{}}$$
2
Where bx2 is the shorter or minor elliptical diameter at exit side (mm). ax2 is the longer or major elliptical diameter at exit side (mm).
To obtain the average roundness of the drilled hole from Eq. (1) and Eq. (2), the Average Roundness Ravg after calculation is obtained from Eq. (3)
$${{R}}_{{a}{v}{g}}^{}=\frac{({{E}}_{{e}{x}}^{}+{{E}}_{{e}{n}}^{})}{2}$$
3
Where Eex is the elliptical roundness at exit Side, Een is the elliptical roundness at entrance side.
When the maximal calculated value of average roundness is 1, it indicates the best quality. For an ideal circle, the larger the roundness, the better the quality.
3 − 2 The Hillock ratio
To evaluate the Hillock ratio of laser drilled holes, irregular holes are considered as a Hillock width shape. The Hillock width is calculated from the inner and outer diameter at entrance side (Hillen) and exit side (Hillex) which is shown in Fig. 3. When measurement of a drill a hole is performed at entrance side, the outer diameter is Oden, and inner diameter is Iden Both Oden and Iden measurements were performed in direction of 0°~180° and 90° ~270°, and the average value was calculated. From the measured data, the hillock ratio Hillen at the entrance side is calculated with Eq. (4). The Hillock ratio at entrance side is obtained by applying Eq. (4).
$${{H}{i}{l}{l}}_{{e}{n}}=\left(\frac{{{O}}_{{d}{e}{n}}^{}-{{I}}_{{d}{e}{n}}^{}}{ {{O}}_{{d}{e}{n}}^{}}\right)$$
4
Where Oden is the outer diameter at entrance side, Iden is the inner diameter at entrance side.
Similarly, measurement at the exit side shows that the outer diameter is Oden, and inner diameter is Idex. Both the Odex and Idex measurements were performed in direction of 0°~180°and 90° ~270°, and the average value was calculated from the measured data. The Hillock ratio at exit side is obtained by applying Eq. (5).
$${{H}{i}{l}{l}}_{{e}{x}}=\left(\frac{{{O}}_{{d}{e}{x}}^{}-{{I}}_{{d}{e}{x}}^{}}{ {{O}}_{{d}{e}{x}}^{}}\right)$$
5
Where Odex is the outer diameter at exit side, and Idex is the inner diameter at exit side.
To obtain the average Hillock quality, the average Hillock ratio (Hillavg) is calculated by applying Eq. (6).
$${{H}{i}{l}{l}}_{{a}{v}{g}} =\left(\frac{{{H}{i}{l}{l}}_{{e}{n}}^{}+{{H}{i}{l}{l}}_{{e}{x}}^{}}{2}\right)$$
6
Where Hillen is the Hillock ratio at entrance side, and Hillex is the Hillock ratio at exit side. When the minimal calculated value of average Hillock ratio is zero, it indicates the best quality. The smaller the hillock ratio, the better the drill quality.
3–3 The taper quality
To evaluate the taper quality, the schematic of taper for a drilled hole is shown in Fig. 4. The taper is defined with Eq. (7).
$${{T}}_{{t}{a}{p}}=\left(\frac{{{I}}_{{d}{e}{n}}-{{I}}_{{d}{e}{x}}}{2．{S}}\right)$$
7
Where Iden is inner diameter at entrance side, and Idex is the inner diameter at exit side. S in this study is the thickness (3 mm) [11].
3–4 The HAZ (Heat-affected zone)
To evaluate the HAZ, the schematic is shown in Fig. 5. The HAZ measurements were performed in direction of 0°~180°and 90° ~270°, and the average value was calculated from the entrance and exit sides. The overall HAZ is obtained by applying Eq. (8).
$${{H}{A}{Z}}_{}^{}=\frac{{({{H}{a}{z}}_{{e}{n}}^{}+{H}{a}{z}}_{{e}{x}}^{})}{2}$$
8
Where Hazen is the average HAZ at entrance side (mm), and Hazex is the average HAZ at exit side (mm)
3–5 Grey relational Generation
Notably, GRA utilizes the mathematical method for analyzing correlations between series comprising a grey relational system, and thereby determines the difference in contribution between a reference series and each compared series. The compared series are alternative vectors created from sets based on attribute characteristics, which are the larger-the-better and the smaller-the-better, or optimization of specific values between the maximum and minimum values of an attribute. Applying a GRA algorithm can rank different alternatives by determining their grey relational grades. The grey relational grades of different series can be used to rank various alternatives, where higher values indicate superior alternatives [24] [27].
Data pre-processing is the first step in the procedure for using GRA. Data pre-processing involves transforming an original sequence into a comparable sequence. Experimental results are thus normalized in a range of 0–1. Eq. (9) shows the calculations for the larger-the-better case; Eq. (10) shows those for the smaller-the-better case; Eq. (11) shows those for the case in which a definite target value must be achieved [24].
$${\text{X}}_{\text{i}}^{\text{＊}}\left(\mathbf{k}\right)=\frac{{\text{X}}_{\text{i}}^{\left(0\right)}\left(\mathbf{k}\right)-\text{m}\text{i}\text{n}\text{all (i)}{\text{X}}_{\text{i}}^{\left(0\right)}\left(\text{k}\right)}{\text{m}\text{a}\text{x}\text{all (i)}{\text{X}}_{\text{i}}^{\left(0\right)}\left(\text{k}\right){\text{X}}_{\text{i}}^{\left(0\right)}\left(\text{k}\right)-\text{m}\text{i}\text{n}\text{all (i)}{\text{X}}_{\text{i}}^{\left(0\right)}\left(\text{k}\right)}$$
9
$${X}_{i}^{＊}\left({k}\right)=\frac{max\text{all (i)}{X}_{i}^{\left(0\right)}\left(k\right)-{X}_{i}^{\left(0\right)}\left(k\right)}{max\text{all (i)}{X}_{i}^{\left(0\right)}\left(k\right)-min\text{all (i)}{X}_{i}^{\left(0\right)}\left(k\right)}$$
10
$${X}_{i}^{＊}\left({k}\right)=1-\frac{\left|{X}_{i}^{\left(0\right)}\right(k)-{O}{B}|}{max\text{all (i)}\{O\text{1},O\text{2}\}}$$
11
Where\({X}_{i}^{＊}\)(k) is the ith grey datum following grey generation for experiment k, xi (0)(k) is the original ith quality datum of experiment k, maxall(i)\({X}_{i}^{\left(0\right)}\)(k) is the maximal value in original sequence, minall(i)\({X}_{i}^{\left(0\right)}\)(k) is the minimal value in the original sequence, OB is the target value, O1 = maxall(i)\({X}_{i}^{\left(0\right)}\)(k)- OB, and O2 = OB- minall(i)\({X}_{i}^{\left(0\right)}\)(k).
The roundness is obtained by applying Eq. (9) Larger-The-Better (LTB). It means that the bigger is the better. The Hillock and HAZ result in minimal value are 0, which indicates the best quality. GRG calculation of HAZ and Hillock are obtained by applying Eq. (10) is the form of Smaller-The-Better (STB). It means that the smaller is the better. The value of taper expects the entrance diameter to be equal to exit diameter. GRG calculation of taper quality which is obtained by applying Eq. (11) is in the form of Nominal-The-Best (NTB), hoping that the zero value is the best among the maximal positive and the minimal negative.
3–6 Grey relational coefficient and grey relational grade
Following data pre-processing, a grey relational coefficient is calculated to express the relationship between ideal and actual normalized experimental results. The grey relational coefficient for four qualities in nine runs is calculated by applying Eq. (12).
\({\gamma }_{i}^{}\left(k\right)=\frac{\varDelta \text{min}\text{ }+{\zeta \varDelta \text{max}}_{}^{}}{\varDelta \text{i}\left(k\right)+\zeta \varDelta \text{max}}\) k = 1, 2,9…. ; i = 1,2,3,4 (12)
$$\varDelta i\left(k\right)=\left|\right|{X}_{i}^{*}\left(0\right)-{X}_{i}^{*}\left({k}\right) \left|\right|$$
13
$$\varDelta \text{max}\text{ }=\genfrac{}{}{0pt}{}{{m}{a}{x}}{\forall {j}\in {i}}\genfrac{}{}{0pt}{}{{m}{a}{x}}{\forall {k}}\left|\right|{X}_{i}^{*}\left({k}\right)-{X}_{j}^{*}\left({k}\right)\left|\right|$$
14
$$\varDelta \text{min}\text{ }=\genfrac{}{}{0pt}{}{{m}{a}{x}}{\forall {j}\in {i}}\genfrac{}{}{0pt}{}{{m}{a}{x}}{\forall {k}}\left|\right|{X}_{i}^{*}\left({k}\right)-{X}_{j}^{*}\left({k}\right)\left|\right|$$
15
Where Δ0i(k) is the deviation sequence between the reference sequence\({X}_{i}^{\text{＊}}\)(0) and \({X}_{i}^{\text{＊}}\)(k), and ζ is the distinguishing or identification coefficient (ζ ε [0,1]) [13] which is set to 0.5 in general case.
However, the importance of each relational coefficient to the final quality is not the same for system requirement. If the weighting factors are not equal, the overall Over all Grey relational grade can be obtained by applying Eq. (16).
\(\gamma \left({k}\right)={\sum }_{i=1}^{4} {\gamma }_{i}^{}\left(k\right)*{W}_{i}^{}\) k= 1, 2,……9 (16)
Where γ(K) is the grey relational grade of each experiment obtained by taking the average of the weighting grey relational coefficients, where wi is the weighting factors for each quality and i = 1,2,3,4 is the quality number. When the gray correlation coefficient is obtained for each quality, the overall grey relational grade quality may be affected by the weights of each equality. The grey relational grade shows the important relationships among the sequences and indicates their degree of influence. The average response of each level effect for All parameters in all runs in Table 2 is discussed as well. The best combination of levels for each parameter can hence be obtained. A detailed description will be given in section 4.