1 Introduction
2 Methodology and experimental design
2.1 Taguchi method
2.2 Principle of GRA
2.3 Principle of KPCA
2.4 Design of experiment and test results
Process parameter | Symbol | Level 1 | Level 2 | Level 3 | Level 4 |
---|---|---|---|---|---|
Type of inserts |
A
| Nose 1 | Nose 2* | ||
Feed rate/ (\(\text{mm}\cdot\text{r}^{ - 1}\)) |
B
| 0.10 | 0.15* | 0.20 | 0.25 |
Depth of cut/ mm |
C
| 0.5 | 1.0* | 1.5 | 2.0 |
Run No. | Factors and their levels | Fy /N | P/W |
μ
| ||
---|---|---|---|---|---|---|
A
|
B
|
C
| ||||
1 | 1 | 1 | 1 | 160.09 | 85.80 | 2.20 |
2 | 1 | 2 | 2 | 225.33 | 208.49 | 2.12 |
3 | 1 | 3 | 3 | 245.72 | 409.87 | 1.92 |
4 | 1 | 4 | 4 | 274.12 | 577.59 | 2.03 |
5 | 1 | 1 | 2 | 177.10 | 200.72 | 1.75 |
6 | 1 | 2 | 1 | 188.69 | 101.66 | 2.58 |
7 | 1 | 3 | 4 | 253.26 | 553.70 | 1.84 |
8 | 1 | 4 | 3 | 266.04 | 428.02 | 2.15 |
9 | 2 | 1 | 3 | 169.67 | 341.47 | 1.46 |
10 | 2 | 2 | 4 | 229.23 | 507.53 | 1.64 |
11 | 2 | 3 | 1 | 200.63 | 107.58 | 2.89 |
12 | 2 | 4 | 2 | 245.34 | 222.39 | 2.97 |
13 | 2 | 1 | 4 | 181.15 | 481.27 | 1.37 |
14 | 2 | 2 | 3 | 206.84 | 358.48 | 1.81 |
15 | 2 | 3 | 2 | 223.52 | 215.21 | 2.52 |
16 | 2 | 4 | 1 | 193.54 | 101.12 | 3.43 |
3 Results and discussion
3.1 Detailed optimization procedure
Run No. | S/N ratios for different responses | Normalization for different responses | ||||
---|---|---|---|---|---|---|
F
y
|
P
|
μ
|
F
y
|
P
|
μ
| |
1 | −44.09 | −39.67 | −6.85 | 1.000 0 | 1.000 0 | 0.483 5 |
2 | −47.06 | −46.38 | −6.52 | 0.364 5 | 0.534 4 | 0.525 0 |
3 | −47.81 | −52.25 | −5.67 | 0.203 4 | 0.179 9 | 0.632 0 |
4 | −48.76 | −55.23 | −6.13 | 0.000 0 | 0.000 0 | 0.574 5 |
5 | −44.96 | −46.05 | −4.84 | 0.812 3 | 0.554 3 | 0.736 1 |
6 | −45.52 | −40.14 | −8.24 | 0.694 4 | 0.911 0 | 0.309 5 |
7 | −48.07 | −54.87 | −5.28 | 0.147 2 | 0.022 2 | 0.681 6 |
8 | −48.50 | −52.63 | −6.64 | 0.055 7 | 0.157 2 | 0.510 3 |
9 | −44.59 | −50.67 | −3.30 | 0.892 0 | 0.275 6 | 0.930 6 |
10 | −47.21 | −54.11 | −4.28 | 0.332 5 | 0.067 8 | 0.807 4 |
11 | −46.05 | −40.63 | −9.20 | 0.580 3 | 0.881 3 | 0.188 3 |
12 | −47.80 | −46.94 | −9.46 | 0.206 3 | 0.500 5 | 0.156 5 |
13 | −45.16 | −53.65 | −2.74 | 0.770 3 | 0.095 7 | 1.000 0 |
14 | −46.31 | −51.09 | −5.14 | 0.523 7 | 0.250 2 | 0.698 5 |
15 | −46.99 | −46.66 | −8.05 | 0.379 5 | 0.517 7 | 0.333 8 |
16 | −45.74 | −40.10 | −10.07 | 0.647 2 | 0.913 8 | 0.000 0 |
Reference sequence | 1.000 0 | 1.000 0 | 1.000 0 |
Run No. | Deviation sequences | Grey relational coefficient |
\(\gamma\)
| Ranking | ||||
---|---|---|---|---|---|---|---|---|
Δoi(Fy) | Δoi(P) | Δoi(μ) |
\(\xi \left( {F_{y} } \right)\)
|
\(\xi \left( P \right)\)
|
\(\xi \left( \mu \right)\)
| |||
1 | 0.000 0 | 0.000 0 | 0.516 5 | 1.000 0 | 1.000 0 | 0.491 9 | 0.830 6 | 1 |
2 | 0.635 5 | 0.465 6 | 0.475 0 | 0.440 3 | 0.517 8 | 0.512 8 | 0.490 3 | 10 |
3 | 0.796 6 | 0.820 1 | 0.368 0 | 0.385 6 | 0.378 8 | 0.576 0 | 0.446 8 | 12 |
4 | 1.000 0 | 1.000 0 | 0.425 5 | 0.333 3 | 0.333 3 | 0.540 3 | 0.402 3 | 16 |
5 | 0.187 7 | 0.445 7 | 0.263 9 | 0.727 1 | 0.528 7 | 0.654 5 | 0.636 8 | 4 |
6 | 0.305 6 | 0.089 0 | 0.690 5 | 0.620 6 | 0.848 9 | 0.420 0 | 0.629 9 | 5 |
7 | 0.852 8 | 0.977 8 | 0.318 4 | 0.369 6 | 0.338 3 | 0.610 9 | 0.439 6 | 13 |
8 | 0.944 3 | 0.842 8 | 0.489 7 | 0.346 2 | 0.372 3 | 0.505 2 | 0.407 9 | 15 |
9 | 0.108 0 | 0.724 4 | 0.069 4 | 0.822 4 | 0.408 4 | 0.878 2 | 0.703 0 | 2 |
10 | 0.667 5 | 0.932 2 | 0.192 6 | 0.428 3 | 0.349 1 | 0.721 9 | 0.499 8 | 9 |
11 | 0.419 7 | 0.118 7 | 0.811 7 | 0.543 7 | 0.808 2 | 0.381 2 | 0.577 7 | 7 |
12 | 0.793 7 | 0.499 5 | 0.843 5 | 0.386 5 | 0.500 3 | 0.372 2 | 0.419 6 | 14 |
13 | 0.229 7 | 0.904 3 | 0.000 0 | 0.685 2 | 0.356 0 | 1.000 0 | 0.680 4 | 3 |
14 | 0.476 3 | 0.749 8 | 0.301 5 | 0.512 1 | 0.400 0 | 0.623 8 | 0.512 0 | 8 |
15 | 0.620 5 | 0.482 3 | 0.666 2 | 0.446 2 | 0.509 0 | 0.428 7 | 0.461 3 | 11 |
16 | 0.352 8 | 0.086 2 | 1.000 0 | 0.586 3 | 0.853 0 | 0.333 3 | 0.590 9 | 6 |
Run No. | Kernel principal components |
\(\gamma_{k}\)
| Ranking | ||
---|---|---|---|---|---|
First | Second | Third | |||
1 | 0.285 3 | − 0.092 4 | − 0.008 8 | 0.206 3 | 1 |
2 | − 0.018 1 | 0.042 8 | 0.001 6 | 0.027 2 | 16 |
3 | − 0.094 4 | 0.036 1 | − 0.001 1 | 0.070 3 | 11 |
4 | − 0.114 7 | 0.057 9 | − 0.003 4 | 0.090 8 | 6 |
5 | 0.021 4 | − 0.074 7 | − 0.002 8 | 0.041 4 | 14 |
6 | 0.169 7 | 0.023 4 | 0.005 2 | 0.110 7 | 4 |
7 | − 0.118 6 | 0.029 3 | − 0.001 3 | 0.082 2 | 8 |
8 | − 0.091 6 | 0.065 4 | − 0.003 0 | 0.079 8 | 10 |
9 | − 0.044 1 | − 0.164 2 | − 0.001 1 | 0.089 1 | 7 |
10 | − 0.122 5 | − 0.017 0 | 0.002 2 | 0.079 9 | 9 |
11 | 0.144 1 | 0.053 7 | 0.006 0 | 0.106 9 | 5 |
12 | − 0.010 0 | 0.093 6 | − 0.002 2 | 0.041 7 | 13 |
13 | − 0.111 1 | − 0.161 9 | 0.007 2 | 0.128 4 | 2 |
14 | − 0.069 6 | − 0.010 2 | − 0.001 5 | 0.045 6 | 12 |
15 | − 0.004 7 | 0.063 8 | − 0.001 7 | 0.027 2 | 15 |
16 | 0.178 8 | 0.054 5 | 0.004 7 | 0.128 0 | 3 |
3.2 Effect of process parameters on the integrated optimization criterion
Experimental factors | Average γ | Range | Ranking | |||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 4 | |||
Type of inserts A | 0.535 5 | 0.555 6* | 0.020 1 | 3 | ||
Feed rate B | 0.712 7* | 0.533 0 | 0.481 4 | 0.455 2 | 0.257 5 | 1 |
Depth of cut C | 0.657 3* | 0.502 0 | 0.517 4 | 0.505 5 | 0.155 3 | 2 |
Experimental factors | Average \(\gamma_{k}\) | Range | Ranking | |||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 4 | |||
Type of inserts A | 0.088 6* | 0.080 8 | 0.007 7 | 3 | ||
Feed rate B | 0.116 3* | 0.065 8 | 0.071 7 | 0.085 1 | 0.050 5 | 2 |
Depth of cut C | 0.137 9* | 0.034 4 | 0.071 2 | 0.095 3 | 0.103 6 | 1 |
3.3 Effect of nonlinearity on multi-response optimization results
Principal components | Eigenvalues | Contribution rate/% |
---|---|---|
First | 0.071 7 | 60.76 |
Second | 0.044 7 | 37.83 |
Third | 0.001 7 | 1.41 |
Performance index | Eigenvectors | ||
---|---|---|---|
First principal component | Second principal component | Third principal component | |
Radial thrust | 0.443 8 | 0.696 5 | −0.563 8 |
Cutting power | 0.814 4 | −0.051 0 | 0.578 0 |
Coefficient of friction | −0.373 8 | 0.715 7 | 0.589 9 |
Performance index | Contribution rate/% | Weight |
---|---|---|
Radial thrust | 19.70 | 0.197 0 |
Cutting power | 66.33 | 0.663 3 |
Coefficient of friction | 13.97 | 0.139 7 |
Experimental factors | Average \(\gamma_{\text{p}}\) | Range | Rank | |||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 4 | |||
Type of inserts A | 0.537 3 | 0.538 3* | 0.001 0 | 3 | ||
Feed rate B | 0.645 2* | 0.529 0 | 0.493 0 | 0.484 0 | 0.161 2 | 2 |
Depth of cut C | 0.774 3* | 0.508 1 | 0.450 6 | 0.418 1 | 0.356 2 | 1 |
3.4 Verification tests
Combinations | Symbol | Radial thrust | Power consumption | Coefficient of friction |
---|---|---|---|---|
Initial |
A
2
B
2
C
2
| 204.17 | 206.51 | 2.21 |
Optimized by \({\gamma \mathord{\left/ {\vphantom {\gamma {\gamma_{\text{p}} }}} \right. \kern-0pt} {\gamma_{\text{p}} }}\) |
A
2
B
1
C
1
| 163.12 | 88.03 | 2.17 |
Relative increment | −20.11% | −57.37% | −1.67% | |
Optimized by \(\gamma_{k}\) |
A
1
B
1
C
1
| 160.09 | 85.80 | 2.20 |
Relative increment | −21.59% | −58.45% | −0.25% |