I have dedicated my research to non-circular gear cutting using wire electrical discharge machining (WEDM). Non-circular gears are special transmission components that can provide a periodically varying transmission ratio, similar to cams and linkages, while retaining the advantages of ordinary gears such as high transmission accuracy, good dynamic performance, reliable operation, and high-speed capability. They are widely used in vehicles, mechanical presses, textile machinery, hydraulic engines, pumps, flowmeters, clocks, and automatic toys. However, the complex pitch curves of non-circular gears make gear cutting very difficult. Traditional gear cutting methods such as hobbing and shaping have inherent limitations: hobbing cannot machine internal gears or concave pitch curves, and shaping requires a specific cutter module, which increases cost. Therefore, I selected WEDM as the gear cutting process because it can machine almost any conductive material regardless of hardness, does not exert mechanical cutting forces, and can produce complex contours in a single setup.
The WEDM process removes material by a series of discrete electrical sparks between a thin wire electrode and the workpiece in a dielectric fluid. Each spark generates a high-temperature plasma channel that melts and vaporizes a small amount of material, leaving a crater. The accumulation of many overlapping craters forms the machined surface. This mechanism is fundamentally different from conventional gear cutting, which relies on mechanical shear. Consequently, the surface integrity produced by WEDM is also different, exhibiting a recast layer, spherical droplets, micro-pores, and an isotropic texture. In my work, I investigated how the key WEDM parameters affect the surface integrity of non-circular gears and how to optimize the gear cutting process.
I designed a Pascal spiral gear as the representative non-circular gear. The pitch curve of a Pascal spiral gear in polar coordinates is given by:
$$
\rho = a + b \cos\theta
$$
where \(a\) and \(b\) are parameters that must satisfy the closure condition:
$$
\int_{0}^{2\pi} \sqrt{\rho^{2} + \left(\frac{d\rho}{d\theta}\right)^{2}} \, d\theta = \pi m z
$$
In my design, I selected \(m = 2\) mm, \(z = 40\), \(a = 18.521\) mm, and \(b = 37.688\) mm. I used the equivalent tooth profile method to generate the gear teeth. The curvature radius at each equivalent tooth position is calculated as:
$$
\gamma = \frac{\left(a^{2} + b^{2} + 2ab\cos\theta\right)^{3/2}}{a^{2} + b^{2} + 3ab\cos\theta}
$$
I divided the pitch curve into 160 equal segments and computed the corresponding angles and curvature radii. Then, I constructed the complete tooth profile. The gear cutting operation was performed on a DK7745 high-speed WEDM machine. I used a molybdenum wire of 0.18 mm diameter as the electrode, deionized water as the dielectric, and a 45 steel plate of 16 mm thickness as the workpiece. The chemical composition and hardness of the 45 steel are listed in Table 1.
| C | Si | Mn | Cr | Ni | Cu | Fe | Microhardness |
|---|---|---|---|---|---|---|---|
| 0.42–0.50 | 0.17–0.37 | 0.50–0.80 | ≤0.25 | ≤0.25 | ≤0.25 | Bal. | 195 HV |
To study the effect of gear cutting parameters, I adopted an orthogonal array L16(4^5) with four factors and four levels. The factors were peak current (A), pulse width (B), pulse interval (C), and tracking (D). The factor levels are shown in Table 2, and the full orthogonal experimental plan is given in Table 3.
| Level | Peak current (A) | Pulse width (μs) | Pulse interval (μs) | Tracking (HZ/s) |
|---|---|---|---|---|
| 1 | 1 | 8 | 5 | 300 |
| 2 | 2 | 16 | 6 | 350 |
| 3 | 3 | 24 | 7 | 400 |
| 4 | 4 | 32 | 8 | 450 |
| Run | A | B | C | D |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 2 | 2 | 2 |
| 3 | 1 | 3 | 3 | 3 |
| 4 | 1 | 4 | 4 | 4 |
| 5 | 2 | 1 | 2 | 3 |
| 6 | 2 | 2 | 1 | 4 |
| 7 | 2 | 3 | 4 | 1 |
| 8 | 2 | 4 | 3 | 2 |
| 9 | 3 | 1 | 3 | 4 |
| 10 | 3 | 2 | 4 | 3 |
| 11 | 3 | 3 | 1 | 2 |
| 12 | 3 | 4 | 2 | 1 |
| 13 | 4 | 1 | 4 | 2 |
| 14 | 4 | 2 | 3 | 1 |
| 15 | 4 | 3 | 2 | 4 |
| 16 | 4 | 4 | 1 | 3 |
After gear cutting, I measured three surface integrity indicators: surface roughness \(R_a\), surface residual stress, and surface microhardness. I selected these indicators because they are the most significant contributors to overall surface integrity in WEDM. To determine their relative importance, I built an analytic hierarchy process (AHP) model. The goal layer was surface integrity of the non-circular gear. The criterion layer included strength, fatigue properties, friction, and corrosion resistance. The indicator layer included surface roughness, surface texture, microhardness, residual stress, microcracks, and other features. The AHP structure is summarized in the weight table, Table 4.
| Indicator | Strength (0.535) | Fatigue (0.270) | Friction (0.120) | Corrosion (0.075) | Overall weight |
|---|---|---|---|---|---|
| Surface roughness | 0.057 | 0.254 | 0.454 | 0.407 | 0.184 |
| Surface texture | 0.057 | 0.162 | 0.308 | 0.194 | 0.126 |
| Microhardness | 0.191 | 0.082 | 0.069 | 0.056 | 0.136 |
| Residual stress | 0.260 | 0.090 | 0.000 | 0.081 | 0.169 |
| Microcracks | 0.396 | 0.372 | 0.131 | 0.237 | 0.346 |
| Other features | 0.041 | 0.041 | 0.038 | 0.025 | 0.039 |
For the AHP, I constructed pairwise comparison matrices. The consistency ratio \(R_c\) was calculated as:
$$
R_c = \frac{I_c}{I_{sc}}
$$
where \(I_c = \frac{\lambda_{\max} – m}{m – 1}\) is the consistency index, and \(I_{sc}\) is the average random consistency index. All matrices had \(R_c < 0.1\), indicating satisfactory consistency. The overall weights in Table 4 were obtained by multiplying the criterion weights with the indicator weights.
Because microcracks had the highest weight, I initially planned to measure them. However, under the selected finishing conditions, the discharge energy was relatively low, and the 45 steel had high thermal conductivity. Therefore, no microcracks were observed. I then focused on the three indicators with the next highest weights: surface roughness, residual stress, and microhardness. I measured surface roughness with a focus variation 3D surface measurement instrument, residual stress with X-ray diffraction using the \(\sin^2\psi\) method, and microhardness with a Vickers microhardness tester. The X-ray diffraction method calculates residual stress as:
$$
\sigma = \frac{E}{(1+\nu)d_0} \left( \frac{d_\psi – d_0}{\sin^2\psi} \right)
$$
where \(E\) is Young’s modulus, \(\nu\) is Poisson’s ratio, \(d_0\) is the stress-free lattice spacing, and \(d_\psi\) is the lattice spacing at tilt angle \(\psi\). The measurement parameters are given in Table 5, and the measured results are shown in Table 6.
| Parameter | Value |
|---|---|
| 2θ range | 120°–180° |
| Step size | 0.02° |
| Speed | 4°/min |
| ψ angles | 10°, 20°, 30°, 45° |
| Run | A | B | C | D | Ra (μm) | Residual stress (MPa) | Microhardness (HV) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 3.7545 | 95.195 | 356.40 |
| 2 | 1 | 2 | 2 | 2 | 3.9214 | 101.106 | 364.60 |
| 3 | 1 | 3 | 3 | 3 | 3.7459 | 107.491 | 375.00 |
| 4 | 1 | 4 | 4 | 4 | 3.6147 | 114.351 | 382.70 |
| 5 | 2 | 1 | 2 | 3 | 2.7551 | 99.780 | 368.00 |
| 6 | 2 | 2 | 1 | 4 | 3.0905 | 106.234 | 377.78 |
| 7 | 2 | 3 | 4 | 1 | 4.2163 | 114.491 | 391.95 |
| 8 | 2 | 4 | 3 | 2 | 5.5580 | 121.351 | 397.50 |
| 9 | 3 | 1 | 3 | 4 | 3.1637 | 105.005 | 381.50 |
| 10 | 3 | 2 | 4 | 3 | 4.2798 | 110.456 | 395.70 |
| 11 | 3 | 3 | 1 | 2 | 4.9394 | 117.381 | 415.10 |
| 12 | 3 | 4 | 2 | 1 | 6.0600 | 147.781 | 418.70 |
| 13 | 4 | 1 | 4 | 2 | 3.6672 | 127.405 | 390.30 |
| 14 | 4 | 2 | 3 | 1 | 4.1078 | 156.856 | 396.80 |
| 15 | 4 | 3 | 2 | 4 | 5.0015 | 139.700 | 422.60 |
| 16 | 4 | 4 | 1 | 3 | 6.3500 | 170.101 | 432.20 |
For parameter optimization, I first converted the measured values into signal-to-noise ratios (SNR). Since surface roughness and residual stress are smaller-the-better characteristics, I used:
$$
\text{SNR} = -10 \log_{10}\left(\frac{1}{n}\sum_{i=1}^{n} y_i^2\right)
$$
Microhardness is a larger-the-better characteristic, so I used:
$$
\text{SNR} = -10 \log_{10}\left(\frac{1}{n}\sum_{i=1}^{n} \frac{1}{y_i^2}\right)
$$
where \(y_i\) is the measured value and \(n\) is the number of measurements. The SNR response tables for the three indicators are given in Tables 7, 8, and 9. From these tables, I found the significance order of the gear cutting parameters. For surface roughness, the order was pulse width > peak current > tracking > pulse interval. For residual stress, the order was peak current > pulse width > tracking > pulse interval. For microhardness, the order was peak current > pulse width > pulse interval > tracking.
| Level | Peak current | Pulse width | Pulse interval | Tracking |
|---|---|---|---|---|
| 1 | -11.49 | -10.40 | -12.81 | -12.97 |
| 2 | -11.50 | -11.64 | -12.58 | -12.98 |
| 3 | -13.04 | -12.96 | -12.16 | -12.24 |
| 4 | -13.40 | -14.44 | -11.89 | -11.24 |
| Range | 1.90 | 4.05 | 0.91 | 1.74 |
| Rank | 2 | 1 | 4 | 3 |
| Level | Peak current | Pulse width | Pulse interval | Tracking |
|---|---|---|---|---|
| 1 | -40.37 | -40.52 | -41.53 | -42.01 |
| 2 | -40.84 | -41.35 | -41.59 | -41.32 |
| 3 | -41.52 | -41.52 | -41.66 | -41.52 |
| 4 | -43.38 | -42.71 | -41.33 | -41.25 |
| Range | 3.02 | 2.19 | 0.33 | 0.76 |
| Rank | 1 | 2 | 4 | 3 |
| Level | Peak current | Pulse width | Pulse interval | Tracking |
|---|---|---|---|---|
| 1 | 51.35 | 51.45 | 51.92 | 51.83 |
| 2 | 51.68 | 51.67 | 51.88 | 51.85 |
| 3 | 52.09 | 52.06 | 51.77 | 51.86 |
| 4 | 52.26 | 52.20 | 51.82 | 51.84 |
| Range | 0.90 | 0.75 | 0.15 | 0.04 |
| Rank | 1 | 2 | 3 | 4 |
For single-objective optimization, the best parameter combination for surface roughness was A1B1C4D4, for residual stress it was A1B1C4D4, and for microhardness it was A4B4C1D3. These combinations conflict with each other, so I needed a multi-objective optimization method. I used fuzzy comprehensive evaluation (FCE) to convert the multiple objectives into a single overall surface integrity index. The FCE process is as follows. First, I defined the factor set \(U = [\text{surface roughness}, \text{residual stress}, \text{microhardness}]\). The evaluation set was \(V = [\text{very poor}, \text{poor}, \text{average}, \text{good}, \text{very good}]\). The weights of the three indicators were determined by AHP as \(A = [0.4934, 0.3108, 0.1958]\). I used a trapezoidal membership function to construct the fuzzy evaluation matrix \(R\) for each experimental run. The membership function for each evaluation level is:
$$
r_{ij} =
\begin{cases}
1, & x_j \le a_1 \\
\frac{x_2 – x_j}{x_2 – x_1}, & a_1 < x_j \le x_2 \\
\frac{x_j – x_1}{x_2 – x_1}, & x_2 < x_j \le a_2 \\
0, & x_j > a_2
\end{cases}
$$
Then, I performed the fuzzy comprehensive operation using the weighted average model:
$$
B = A \circ R
$$
Finally, I computed a single surface integrity value (SI) using the weighted membership principle:
$$
\text{SI} = \frac{\sum_{n=2}^{5} b_n^2 \cdot n}{\sum_{n=2}^{5} b_n^2}
$$
The SI values for all runs are listed in Table 10. A larger SI indicates better surface integrity. The range analysis of SI is shown in Table 11. The significance order of the gear cutting parameters was pulse width > peak current > tracking > pulse interval. The optimal parameter combination was A2B1C4D4, which means peak current 2 A, pulse width 8 μs, pulse interval 8 μs, and tracking 450 HZ/s.
| Run | SI |
|---|---|
| 1 | 2.394 |
| 2 | 2.083 |
| 3 | 2.308 |
| 4 | 2.307 |
| 5 | 2.788 |
| 6 | 2.575 |
| 7 | 2.112 |
| 8 | 1.697 |
| 9 | 2.559 |
| 10 | 2.219 |
| 11 | 1.991 |
| 12 | 1.376 |
| 13 | 2.189 |
| 14 | 1.859 |
| 15 | 1.753 |
| 16 | 1.143 |
| Level | Peak current | Pulse width | Pulse interval | Tracking |
|---|---|---|---|---|
| 1 | 2.273 | 2.483 | 2.026 | 1.935 |
| 2 | 2.293 | 2.184 | 2.000 | 1.990 |
| 3 | 2.036 | 2.041 | 2.106 | 2.115 |
| 4 | 1.736 | 1.631 | 2.207 | 2.299 |
| Range | 0.557 | 0.852 | 0.207 | 0.363 |
| Rank | 2 | 1 | 4 | 3 |
To validate my FCE-based optimization, I compared it with the widely used grey relational analysis (GRA). I calculated the grey relational coefficients (GRC) and grey relational grades (GRG) for all runs. The GRC is calculated as:
$$
\xi_{ij} = \frac{\min_i \min_j |x_i^0 – x_{ij}| + \rho \max_i \max_j |x_i^0 – x_{ij}|}{|x_i^0 – x_{ij}| + \rho \max_i \max_j |x_i^0 – x_{ij}|}
$$
where \(\rho = 0.5\) is the distinguishing coefficient. The GRG is the average of the GRC values for each run. The GRG values are shown in Table 12. The range analysis of GRG is given in Table 13. The optimal combination from GRA was also A2B1C4D4, which matches my FCE result. This confirms the effectiveness of the FCE method for gear cutting parameter optimization.
| Run | GRC Ra | GRC residual stress | GRC microhardness | GRG |
|---|---|---|---|---|
| 1 | 0.8248 | 0.9132 | 0.6357 | 0.7912 |
| 2 | 0.8050 | 0.8737 | 0.6523 | 0.7770 |
| 3 | 0.8259 | 0.8368 | 0.6739 | 0.7789 |
| 4 | 0.8429 | 0.8026 | 0.6905 | 0.7787 |
| 5 | 1.0000 | 0.8820 | 0.6592 | 0.8471 |
| 6 | 0.9269 | 0.8436 | 0.6799 | 0.8168 |
| 7 | 0.7740 | 0.8019 | 0.7111 | 0.7623 |
| 8 | 0.6750 | 0.7723 | 0.7237 | 0.7237 |
| 9 | 0.9133 | 0.8504 | 0.6879 | 0.8172 |
| 10 | 0.7679 | 0.8214 | 0.7196 | 0.7696 |
| 11 | 0.7140 | 0.7889 | 0.7658 | 0.7562 |
| 12 | 0.6490 | 0.6862 | 0.7748 | 0.7033 |
| 13 | 0.8360 | 0.7490 | 0.7074 | 0.7641 |
| 14 | 0.7849 | 0.6639 | 0.7221 | 0.7236 |
| 15 | 0.7096 | 0.7088 | 0.7847 | 0.7344 |
| 16 | 0.6357 | 0.6357 | 0.8097 | 0.6937 |
| Level | Peak current | Pulse width | Pulse interval | Tracking |
|---|---|---|---|---|
| 1 | 0.781 | 0.805 | 0.764 | 0.745 |
| 2 | 0.787 | 0.772 | 0.765 | 0.755 |
| 3 | 0.762 | 0.758 | 0.761 | 0.772 |
| 4 | 0.729 | 0.725 | 0.769 | 0.787 |
| Range | 0.059 | 0.080 | 0.008 | 0.042 |
| Rank | 2 | 1 | 4 | 3 |
I performed a confirmation experiment using the optimal gear cutting parameters A2B1C4D4. The results are shown in Table 14. Compared with the initial optimal run (run 5), the surface roughness decreased from 2.7551 μm to 1.3840 μm, the residual stress increased slightly from 99.780 MPa to 112.748 MPa, and the microhardness decreased slightly from 368.00 HV to 365.49 HV. The overall surface integrity increased from 2.7880 to 3.3280, an improvement of 19.37%. This confirms that the FCE-based multi-objective optimization is effective for non-circular gear cutting by WEDM.
| Indicator | Initial level (A2B1C2D3) | Optimal level (A2B1C4D4) | Improvement |
|---|---|---|---|
| Surface roughness (μm) | 2.7551 | 1.3840 | — |
| Residual stress (MPa) | 99.7800 | 112.7480 | — |
| Microhardness (HV) | 368.0000 | 365.4880 | — |
| Surface integrity | 2.7880 | 3.3280 | 19.37% |
After optimizing the gear cutting parameters, I also built a prediction model for surface integrity. I used support vector regression (SVR) because it has good generalization ability with small samples. The SVR model for nonlinear regression is:
$$
f(x) = \sum_{i=1}^{n} (\alpha_i – \alpha_i^*) K(x_i, x) + b
$$
where \(K(x_i, x)\) is the kernel function. The SVR optimization problem is:
$$
\min_{\omega, b, \xi, \xi^*} \frac{1}{2} \|\omega\|^2 + C \sum_{i=1}^{n} (\xi_i + \xi_i^*)
$$
subject to:
$$
\begin{aligned}
y_i – \omega^T x_i – b &\le \varepsilon + \xi_i \\
\omega^T x_i + b – y_i &\le \varepsilon + \xi_i^* \\
\xi_i, \xi_i^* &\ge 0
\end{aligned}
$$
I compared three common kernel functions: polynomial, radial basis function (RBF), and sigmoid. Their performances are shown in Table 15. The RBF kernel gave the best results for all three indicators. Therefore, I selected the RBF kernel for the final prediction model.
| Kernel | Ra R² | Ra MAPE | Residual stress R² | Residual stress MAPE | Microhardness R² | Microhardness MAPE |
|---|---|---|---|---|---|---|
| Poly | 0.9624 | 5.4115 | 0.9716 | 2.3867 | 0.9533 | 0.7808 |
| RBF | 0.9967 | 1.2761 | 0.9992 | 0.4151 | 0.9941 | 0.3017 |
| Sigmoid | 0.9347 | 4.9032 | 0.9357 | 3.4287 | 0.9926 | 0.4168 |
To further improve the prediction accuracy, I optimized the SVR parameters using a hybrid particle swarm optimization (HPSO) algorithm. The standard PSO updates particle velocity and position as:
$$
v_{ij}(t+1) = \omega v_{ij}(t) + c_1 r_1 [p_{ij}(t) – x_{ij}(t)] + c_2 r_2 [p_{gj}(t) – x_{ij}(t)]
$$
$$
x_{ij}(t+1) = x_{ij}(t) + v_{ij}(t+1)
$$
I introduced an oscillatory decreasing inertia weight:
$$
\omega(t) = \omega_{\max} – (\omega_{\max} – \omega_{\min}) \frac{t}{T_{\max}} + \frac{1 – \sin(\frac{t}{T_{\max}} \cdot \frac{\pi}{2})}{1 + \sin(\frac{t}{T_{\max}} \cdot \frac{\pi}{2})}
$$
and a trade-off strategy between local and global search inspired by the whale optimization algorithm. The parameter \(A\) is computed as:
$$
a(t) = 2\left(1 – \frac{t}{T_{\max}}\right)
$$
$$
A(t) = a(t)(2r – 1)
$$
When \(|A| < 1\), the algorithm performs local search:
$$
v_{ij}(t+1) = \omega v_{ij}(t) + c_1 r_1 [p_{bj}(t) – x_{ij}(t)] + c_2 r_2 [g_{bj}(t) – x_{ij}(t)]
$$
When \(|A| \ge 1\), the algorithm performs global search:
$$
v_{ij}(t+1) = \omega v_{ij}(t) – x_{rand}(t)
$$
I tested the HPSO on four standard benchmark functions: Sphere, Rosenbrock, Griewank, and Rastrigin. The results are shown in Table 16. The HPSO achieved better convergence accuracy and speed than the standard PSO with concave inertia weight.
| Function | Mean HPSO | Mean PSO | Std HPSO | Std PSO |
|---|---|---|---|---|
| Sphere | -0.01 | -0.1 | 0.000406 | 0.0338 |
| Rosenbrock | -10.12 | -52.49 | 118.94 | 6649.50 |
| Griewank | -0.99948 | -0.99897 | 0.99897 | 0.99906 |
| Rastrigin | -27.22 | -40.74 | 1224.72 | 1766.43 |
Using the HPSO-optimized SVR with RBF kernel, I predicted the three surface integrity indicators. The prediction performance is shown in Table 17. The R² values were all above 0.94, and the MAPE values were low, indicating high prediction accuracy. The predicted values closely matched the experimental values. The R² and MAPE for each indicator were: surface roughness R² = 0.9978, MAPE = 0.4534; residual stress R² = 0.9483, MAPE = 3.1673; microhardness R² = 0.9786, MAPE = 0.4779. These results demonstrate that the HPSO-SVR model can accurately predict the surface integrity of non-circular gears produced by WEDM gear cutting.
| Indicator | R² | MAPE (%) |
|---|---|---|
| Surface roughness | 0.9978 | 0.4534 |
| Residual stress | 0.9483 | 3.1673 |
| Microhardness | 0.9786 | 0.4779 |
Finally, I developed a surface quality prediction system for non-circular gear WEDM. The system was built with Python 3.8.8 and the Tkinter module. It has five main interfaces: login, process indicator prediction, process parameter optimization, data display, and about. The system allows users to predict surface roughness, residual stress, and microhardness from given gear cutting parameters, optimize gear cutting parameters to meet desired surface integrity requirements, and save or export the results. This system can help improve the intelligence level of non-circular gear cutting and reduce reliance on operator experience.
In summary, my research provides a comprehensive study on parameter optimization and surface integrity prediction for non-circular gear cutting by WEDM. The main contributions are: (1) I established a surface integrity evaluation system specifically for WEDM gear cutting using AHP, which is different from conventional cutting evaluation systems. (2) I used SNR and FCE to optimize the gear cutting parameters, and the results were validated by GRA. The optimal parameters were A2B1C4D4, achieving a 19.37% improvement in overall surface integrity. (3) I built an HPSO-SVR prediction model with RBF kernel, achieving high accuracy (R² > 0.94). (4) I developed a practical prediction system for gear cutting engineers. This work contributes to the advancement of non-circular gear cutting by WEDM and provides a basis for intelligent manufacturing of complex gears.
Future work could include more non-electrical parameters such as wire tension and dielectric flushing pressure, a wider range of gear cutting parameters, and performance testing of the optimized gears. The keyword ‘gear cutting’ has been central to my research, and I believe that optimizing gear cutting parameters and predicting surface integrity are essential steps toward high-quality non-circular gear production.