In my work, I studied how wire electrical discharge machining, which I refer to as WEDM, affects the surface integrity of non-circular gears. I focused on a Pascal worm-line gear because this kind of non-circular gear cutting is representative of complex gear cutting tasks in which the pitch curve is not circular and the tooth profile changes along the gear. I treated surface integrity as a multi-parameter system rather than a single roughness number. In my experiments, I selected surface morphology, surface roughness, surface residual stress, and surface microhardness as the main responses. I used a fast-wire WEDM machine, 45 steel, a molybdenum wire, and an emulsion-type working fluid. I combined finite element simulation, orthogonal gear cutting experiments, statistical analysis, weighting methods, grey comprehensive evaluation, machine-learning prediction, and the snake optimizer. Throughout this article, I use the term gear cutting repeatedly because the core question is how WEDM parameters influence gear cutting quality and how I can predict and optimize that quality.
Process Principle of WEDM in Gear Cutting
In gear cutting by WEDM, I connect the workpiece to the positive pole of a pulsed power supply and the moving electrode wire to the negative pole. The dielectric medium is ionized and broken down, a discharge channel forms, and the material is melted or vaporized by the extremely high local temperature. The removed material is flushed away by the working fluid, and the gap is deionized during the pulse-off time. I treat one complete cycle as four stages: breakdown and channel formation, melting and vaporization, ejection of molten material, and deionization. This cycle repeats many times during gear cutting, and the accumulated discharge craters define the final surface integrity of the non-circular gear.
Because the discharge process is transient and highly localized, I modeled the thermal problem as a nonlinear transient heat conduction problem in a cylindrical coordinate system. The governing equation I used is:
$$
\rho c(T)\frac{\partial T}{\partial t}
=
\frac{1}{r}\frac{\partial}{\partial r}\left(\lambda(T) r \frac{\partial T}{\partial r}\right)
+
\frac{\partial}{\partial z}\left(\lambda(T)\frac{\partial T}{\partial z}\right)
+
Q(r,z,t)
$$
Here, \(T\) is temperature, \(t\) is time, \(\rho\) is density, \(c(T)\) is specific heat capacity, \(\lambda(T)\) is thermal conductivity, and \(Q(r,z,t)\) is the heat generated per unit volume and time. For the single-pulse simulation, I made a set of assumptions that are common in WEDM gear cutting research: one discharge channel forms per pulse, the channel is cylindrical, the heat source follows a Gaussian distribution, the domain is axisymmetric, the material is homogeneous and isotropic, voltage and current are stable during the pulse, molten material is fully expelled, and no recast layer remains in the ideal simulation.
Discharge Channel and Heat Source
I used an empirical relation to estimate the discharge channel radius because the real radius during gear cutting is difficult to measure directly. The relation I adopted is:
$$
R(t)=2.04 \times 10^{-3} t^{0.43} I^{0.44}
$$
In this expression, \(I\) is peak current in amperes and \(t\) is pulse-on time in microseconds. I also assumed a Gaussian heat flux distribution, as shown below:
$$
q(r)=\frac{4.57 \eta U I}{\pi R^2(t)}
\exp\left(-4.5\frac{r^2}{R^2(t)}\right)
$$
Here, \(U\) is machining voltage, \(\eta\) is the energy distribution coefficient, and \(R(t)\) is the discharge channel radius. I set the energy distribution coefficient to 0.3 because only part of the total discharge energy enters the positive workpiece during WEDM gear cutting. The boundary condition I applied on the top surface was:
$$
-k\frac{\partial T}{\partial z}=q(r)\quad \text{for } r\leq R
$$
$$
-k\frac{\partial T}{\partial z}=0\quad \text{for } r>R
$$
Outside the discharge zone, I applied convective cooling with the working fluid. The initial temperature was set to room temperature. I used temperature-dependent thermal properties for 45 steel, including thermal conductivity and specific heat capacity, because the discharge temperature in gear cutting by WEDM can exceed the melting point by a large margin.
| Temperature (K) | Thermal conductivity (W/(m2 K)) | Specific heat capacity (J/(kg K)) |
|---|---|---|
| 298 | 48 | 445 |
| 500 | 46 | 529 |
| 700 | 41 | 615 |
| 900 | 35 | 825 |
| 1028 | 25 | 1064 |
| 1100 | 26 | 827 |
| 1500 | 29 | 652 |
| 1809 | 29 | 822 |
| 3135 | 29 | 821 |
Temperature Field Simulation for Gear Cutting
I built a quarter model to reduce computation time. The model size was 120 μm by 120 μm by 100 μm, and I refined the mesh near the discharge center. I used a mesh size of about 5 μm in the refined region. I simulated two groups of conditions. In the first group, I fixed pulse-on time at 16 μs and varied peak current from 1 A to 3 A. In the second group, I fixed peak current at 4 A and varied pulse-on time from 8 μs to 24 μs. I used 70 V as the machining voltage. The simulation matrix is shown below.
| Voltage (V) | Peak current (A) | Pulse-on time (μs) | Purpose in my gear cutting study |
|---|---|---|---|
| 70 | 1 | 16 | low-energy gear cutting |
| 70 | 2 | 16 | medium-energy gear cutting |
| 70 | 3 | 16 | high-energy gear cutting |
| 70 | 4 | 8 | short-pulse gear cutting |
| 70 | 4 | 16 | reference gear cutting |
| 70 | 4 | 24 | long-pulse gear cutting |
The simulation results showed that the maximum temperature increased with peak current when pulse-on time was constant. For pulse-on time of 16 μs, the maximum temperatures were approximately 4285.1 °C, 5924.4 °C, and 7002.4 °C for 1 A, 2 A, and 3 A, respectively. The maximum temperature was located at the center of the discharge zone and decreased rapidly along both radial and depth directions. I attributed this rapid decrease to working-fluid cooling and to the thermal properties of 45 steel. The discharge crater depth also increased with peak current. The simulated crater depths were approximately 5.52 μm, 8.05 μm, and 9.60 μm for 1 A, 2 A, and 3 A, respectively.
When I fixed peak current at 4 A, the maximum temperature decreased as pulse-on time increased. The maximum temperatures were approximately 8955.3 °C, 7890.5 °C, and 7110.3 °C for 8 μs, 16 μs, and 24 μs. I explained this by the increase in discharge channel radius with pulse-on time, which reduced the heat flux density. The crater depth increased with pulse-on time, with values of approximately 8.01 μm, 10.57 μm, and 12.24 μm for 8 μs, 16 μs, and 24 μs. These simulation results helped me understand why peak current and pulse-on time are dominant factors in WEDM gear cutting surface integrity.
Non-Circular Gear Design and Experimental Setup
I selected a Pascal worm-line gear as the non-circular gear for my gear cutting study. I designed the pitch curve and then generated the tooth profile by a profile-shaping method. The main parameters I used were an generating circle diameter of 18.521 mm, a fixed length of 37.688 mm, a module of 2 mm, a tooth number of 40, and a face width of 16 mm. I machined the gear on a DK7745 fast-wire WEDM machine. The maximum cutting thickness of the machine was 400 mm, the cutting speed was above 100 mm2/min, and the mechanical accuracy was sufficient for my gear cutting experiments. I used 45 steel as the workpiece material because it is widely used in mechanical parts and has well-known thermal properties. I used a molybdenum wire with a diameter of 0.18 mm, an emulsion-type working fluid, and a fixed machining voltage of 70 V.
| Item | Selection in my gear cutting experiments |
|---|---|
| Non-circular gear type | Pascal worm-line gear |
| Workpiece material | 45 steel |
| Machine | DK7745 fast-wire WEDM machine |
| Wire electrode | Molybdenum wire, 0.18 mm |
| Working fluid | Emulsion-type working fluid |
| Machining voltage | 70 V |
| Gear module | 2 mm |
| Tooth number | 40 |
| Face width | 16 mm |
For the gear cutting experiments, I selected four factors that I considered most important in WEDM: peak current, pulse-on time, pulse-off time, and tracking. Peak current controls the instantaneous discharge energy. Pulse-on time controls the duration of each discharge. Pulse-off time controls deionization and debris flushing. Tracking controls the gap state and feed regulation. I used four levels for each factor and an L16 orthogonal array. This allowed me to study multiple parameters in a reasonable number of gear cutting runs.
| Level | A: Peak current (A) | B: Pulse-on time (μs) | C: Pulse-off time (μs) | D: Tracking (level) |
|---|---|---|---|---|
| 1 | 1 | 8 | 5 | 2 |
| 2 | 2 | 16 | 6 | 3 |
| 3 | 3 | 24 | 7 | 4 |
| 4 | 4 | 32 | 8 | 5 |
The tracking parameter had preset maximum values for each level. I used these values to maintain stable discharge conditions during gear cutting.
| Tracking level | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Preset maximum (Hz/s) | 250 | 300 | 350 | 400 | 450 | 500 |
Measurement Methods for Surface Integrity
I measured surface morphology and surface roughness with a non-contact optical three-dimensional surface measurement system. I selected three positions on each specimen and averaged the reliable values to reduce measurement error. I measured surface residual stress by X-ray diffraction. This method is nondestructive and suitable for gear cutting surfaces. I cleaned the specimen surface before measurement. I measured surface microhardness with a fully automatic Vickers microhardness tester. I used a load of 9.807 N and a holding time of 15 s. The Vickers hardness was calculated as:
$$
HV=\frac{1854.4 P}{d^2}
$$
In this equation, \(P\) is the applied load in mN and \(d\) is the mean diagonal length of the indentation in μm. I measured the base hardness of 45 steel as approximately 195 HV. After WEDM gear cutting, the surface hardness increased because of the recast layer and the heat-affected layer.
| Response | Instrument | Principle | Key setting |
|---|---|---|---|
| Surface morphology | Optical 3D surface measurement system | Non-contact focus variation | Three measurement positions |
| Surface roughness | Optical 3D surface measurement system | Areal roughness extraction | Average of reliable values |
| Residual stress | X-ray diffractometer | X-ray diffraction | Nondestructive stress measurement |
| Microhardness | Automatic Vickers microhardness tester | Vickers indentation | 9.807 N, 15 s |
Orthogonal Gear Cutting Results
I completed sixteen gear cutting runs with the L16 orthogonal array. The measured responses were surface roughness \(Ra\), surface residual stress \(\sigma\), and surface microhardness \(\mu h\). The results are summarized below. I used these data for variance analysis, range analysis, comprehensive evaluation, prediction, and multi-objective optimization.
| Run | A: Peak current (A) | B: Pulse-on time (μs) | C: Pulse-off time (μs) | D: Tracking | Ra (μm) | σ (MPa) | μh (HV) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 8 | 5 | 2 | 3.75 | 95.1954 | 356.4 |
| 2 | 1 | 16 | 6 | 3 | 3.92 | 101.1058 | 364.6 |
| 3 | 1 | 24 | 7 | 4 | 3.75 | 107.4910 | 375.0 |
| 4 | 1 | 32 | 8 | 5 | 3.61 | 114.3510 | 382.7 |
| 5 | 2 | 8 | 6 | 4 | 2.76 | 99.7800 | 368.0 |
| 6 | 2 | 16 | 5 | 5 | 3.09 | 106.2340 | 377.78 |
| 7 | 2 | 24 | 8 | 2 | 4.22 | 114.4910 | 391.95 |
| 8 | 2 | 32 | 7 | 3 | 5.56 | 121.3510 | 397.5 |
| 9 | 3 | 8 | 7 | 5 | 3.16 | 105.0054 | 381.5 |
| 10 | 3 | 16 | 5 | 4 | 4.28 | 110.4558 | 395.7 |
| 11 | 3 | 24 | 8 | 3 | 4.94 | 117.3809 | 415.1 |
| 12 | 3 | 32 | 6 | 2 | 6.06 | 147.7810 | 418.7 |
| 13 | 4 | 8 | 8 | 3 | 3.67 | 127.4054 | 390.3 |
| 14 | 4 | 16 | 7 | 2 | 4.11 | 156.8558 | 396.8 |
| 15 | 4 | 24 | 6 | 5 | 5.00 | 139.7009 | 422.6 |
| 16 | 4 | 32 | 5 | 4 | 6.35 | 170.1010 | 432.2 |
Surface Morphology After Gear Cutting
I observed that WEDM gear cutting produces a surface composed of irregular craters and re-solidified material. The craters form because the pulse discharge melts and vaporizes the material, and the molten material is then flushed away by the working fluid. The re-solidified material forms when molten material is not fully removed and is cooled by the dielectric. I found that crater size increased with peak current and pulse-on time. When the discharge energy was low, the craters were smaller and shallower, and the surface morphology was better. When the discharge energy was high, more molten material was produced, and more re-solidified material remained on the surface. This is a central reason why gear cutting by WEDM must be controlled carefully, especially for non-circular gears whose teeth vary in curvature and local heat accumulation.
I also noted that surface morphology is not identical along the whole non-circular gear. The local curvature changes, so the effective gap condition and debris evacuation can change during gear cutting. This means that the same nominal parameters can produce slightly different surface integrity at different tooth positions. I accounted for this by measuring multiple positions and by treating the non-circular gear as a complex part whose gear cutting quality must be evaluated by multiple responses.
Analysis of Surface Roughness
I used variance analysis to determine the significance of each factor on surface roughness. At the 95% confidence level, peak current and pulse-on time were highly significant, while pulse-off time and tracking were significant. The order of influence was pulse-on time, peak current, tracking, and pulse-off time. The variance results are shown below.
| Source | Type III sum of squares | Degrees of freedom | Mean square | F value | P value |
|---|---|---|---|---|---|
| Peak current | 3.088 | 3 | 1.029 | 16.033 | 0.024 |
| Pulse-on time | 10.970 | 3 | 3.657 | 56.964 | 0.004 |
| Pulse-off time | 2.093 | 3 | 0.698 | 10.865 | 0.040 |
| Tracking | 2.794 | 3 | 0.931 | 14.508 | 0.027 |
| Error | 0.193 | 3 | 0.064 | — | — |
I then used range analysis to find the best level for surface roughness alone. The range values showed that pulse-on time had the largest effect, followed by peak current, tracking, and pulse-off time. The optimal combination for minimum surface roughness was A1B1C4D4, which means 1 A peak current, 8 μs pulse-on time, 8 μs pulse-off time, and tracking level 5.
| Factor | Peak current | Pulse-on time | Pulse-off time | Tracking |
|---|---|---|---|---|
| K1 average | 3.7591 | 3.3351 | 4.3687 | 4.5347 |
| K2 average | 3.9050 | 3.8499 | 4.4345 | 4.5215 |
| K3 average | 4.6107 | 4.4758 | 4.1439 | 4.2827 |
| K4 average | 4.7816 | 5.3957 | 4.1094 | 3.7176 |
| Range | 1.0225 | 2.0606 | 0.3251 | 0.8171 |
| Best level | 1 | 1 | 4 | 4 |
I found that surface roughness increased quickly when peak current or pulse-on time increased. This agrees with my simulation, because both parameters increase the discharge energy and enlarge the crater. Pulse-off time and tracking also affected roughness, but their influence was smaller. A longer pulse-off time improves debris flushing and gap recovery, while a suitable tracking level stabilizes the discharge. For non-circular gear cutting, this means that if roughness is the only requirement, low energy and stable flushing are preferred.
Analysis of Surface Residual Stress
I measured residual stress because it strongly affects fatigue life and dimensional stability. In my WEDM gear cutting experiments, the surface residual stress was tensile. I used variance analysis to identify significant factors. Peak current and pulse-on time were significant, while pulse-off time and tracking were not significant at the chosen confidence level. The order of influence was peak current, pulse-on time, tracking, and pulse-off time.
| Source | Type III sum of squares | Degrees of freedom | Mean square | F value | P value |
|---|---|---|---|---|---|
| Peak current | 4559.472 | 3 | 1519.824 | 28.792 | 0.010 |
| Pulse-on time | 2038.992 | 3 | 679.664 | 12.876 | 0.032 |
| Pulse-off time | 35.085 | 3 | 11.695 | 0.222 | 0.876 |
| Tracking | 372.559 | 3 | 124.186 | 2.353 | 0.250 |
| Error | 158.360 | 3 | 52.787 | — | — |
The range analysis for residual stress showed that the optimal combination for minimum tensile residual stress was also A1B1C4D4, namely 1 A peak current, 8 μs pulse-on time, 8 μs pulse-off time, and tracking level 5.
| Factor | Peak current | Pulse-on time | Pulse-off time | Tracking |
|---|---|---|---|---|
| K1 average | 104.54 | 106.85 | 120.49 | 128.58 |
| K2 average | 110.46 | 118.66 | 122.09 | 116.81 |
| K3 average | 120.16 | 119.77 | 122.68 | 121.96 |
| K4 average | 148.52 | 138.39 | 118.41 | 116.32 |
| Range | 43.98 | 31.55 | 4.27 | 12.26 |
| Best level | 1 | 1 | 4 | 4 |
I explain these results by the thermal nature of WEDM gear cutting. The surface can be considered as a thermally treated layer. Thermal stress is the dominant contributor to residual stress, while plastic deformation and phase transformation are secondary. When peak current and pulse-on time increase, the discharge energy increases, the local temperature gradient becomes steeper, and the tensile residual stress becomes larger. A longer pulse-off time gives more time for deionization and debris removal, which improves stability, but it does not remove the thermal origin of the stress. Tracking affects the gap state, but its direct effect on residual stress is weaker than the electrical energy parameters.
Analysis of Surface Microhardness
I measured surface microhardness to characterize the hardened layer produced by WEDM gear cutting. The base hardness of 45 steel was about 195 HV, and all machined surfaces were harder than the base. I attributed this to the recast layer and the heat-affected layer. The variance analysis showed that peak current and pulse-on time were highly significant, while pulse-off time and tracking were less influential. The order of influence was peak current, pulse-on time, pulse-off time, and tracking.
| Source | Type III sum of squares | Degrees of freedom | Mean square | F value | P value |
|---|---|---|---|---|---|
| Peak current | 4087.973 | 3 | 1362.658 | 235.900 | 0.000 |
| Pulse-on time | 2885.970 | 3 | 961.990 | 166.538 | 0.001 |
| Pulse-off time | 143.563 | 3 | 47.854 | 8.284 | 0.058 |
| Tracking | 17.912 | 3 | 5.971 | 1.034 | 0.489 |
| Error | 17.329 | 3 | 5.776 | — | — |
The range analysis for microhardness showed that the best level for maximum hardness was A4B4C4D3, which means 4 A peak current, 32 μs pulse-on time, 8 μs pulse-off time, and tracking level 4.
| Factor | Peak current | Pulse-on time | Pulse-off time | Tracking |
|---|---|---|---|---|
| K1 average | 369.675 | 374.050 | 390.520 | 369.675 |
| K2 average | 383.808 | 383.720 | 393.475 | 383.808 |
| K3 average | 402.750 | 401.163 | 387.700 | 402.750 |
| K4 average | 410.475 | 407.775 | 395.013 | 410.475 |
| Range | 40.80 | 33.725 | 7.313 | 40.80 |
| Best level | 4 | 4 | 4 | 3 |
I observed a conflict between the requirements for minimum roughness, minimum tensile residual stress, and maximum microhardness. A parameter set that reduces roughness and residual stress tends to reduce the hardened layer thickness and microhardness. A parameter set that maximizes microhardness tends to increase discharge energy and thus worsen roughness and residual stress. This conflict is the main reason I moved from single-response analysis to a comprehensive evaluation and multi-objective optimization. In gear cutting by WEDM, a compromise is necessary, and the compromise must be based on the actual service requirements of the non-circular gear.
Data Normalization and Weighting
Because surface roughness, residual stress, and microhardness have different units and different optimization directions, I normalized the data before comprehensive evaluation. I used a min-max normalization method. For a larger-is-better response, I used:
$$
x_{ij}=\frac{y_{ij}-\min_j y_{ij}}{\max_j y_{ij}-\min_j y_{ij}}\times 0.998+0.002
$$
For a smaller-is-better response, I used:
$$
x_{ij}=\frac{\max_j y_{ij}-y_{ij}}{\max_j y_{ij}-\min_j y_{ij}}\times 0.998+0.002
$$
I treated surface roughness and residual stress as smaller-is-better responses, and microhardness as a larger-is-better response. The normalized values are shown below.
| Run | Normalized Ra | Normalized σ | Normalized μh |
|---|---|---|---|
| 1 | 0.7226 | 1.0000 | 0.0020 |
| 2 | 0.6762 | 0.9213 | 0.1100 |
| 3 | 0.7249 | 0.8362 | 0.2469 |
| 4 | 0.7614 | 0.7448 | 0.3483 |
| 5 | 1.0000 | 0.9389 | 0.1547 |
| 6 | 0.9069 | 0.8529 | 0.2835 |
| 7 | 0.5943 | 0.7429 | 0.4701 |
| 8 | 0.2219 | 0.6515 | 0.5431 |
| 9 | 0.8866 | 0.8693 | 0.3325 |
| 10 | 0.5767 | 0.7967 | 0.5194 |
| 11 | 0.3936 | 0.7044 | 0.7749 |
| 12 | 0.0825 | 0.2994 | 0.8223 |
| 13 | 0.7468 | 0.5709 | 0.4483 |
| 14 | 0.6245 | 0.1785 | 0.5339 |
| 15 | 0.3764 | 0.4070 | 0.8736 |
| 16 | 0.0020 | 0.0020 | 1.0000 |
I used both a subjective method and an objective method to determine weights. I used the analytic hierarchy process as the subjective method because I wanted to include engineering judgment about the importance of roughness, residual stress, and microhardness. I used the entropy weight method as the objective method because it extracts information from the data dispersion. I then used a game-theory combination weighting method to balance the two.
For the analytic hierarchy process, I constructed a pairwise comparison matrix. The weight calculation was:
$$
w_i=\frac{\sqrt[n]{\prod_{j=1}^n a_{ij}}}{\sum_{i=1}^n \sqrt[n]{\prod_{j=1}^n a_{ij}}}
$$
I checked consistency with:
$$
CI=\frac{\lambda_{\max}-n}{n-1}
$$
$$
CR=\frac{CI}{RI}
$$
For the entropy weight method, I calculated the proportion, entropy, redundancy, and weight as:
$$
p_{ij}=\frac{x_{ij}}{\sum_{i=1}^n x_{ij}}
$$
$$
e_j=-k\sum_{i=1}^n p_{ij}\ln p_{ij},\quad k=\frac{1}{\ln n}
$$
$$
g_j=1-e_j
$$
$$
w_j=\frac{g_j}{\sum_{j=1}^m g_j}
$$
The weights from the two methods and the combined weights are shown below.
| Method | Ra weight | σ weight | μh weight |
|---|---|---|---|
| AHP | 0.5500 | 0.2402 | 0.2098 |
| Entropy weight | 0.3251 | 0.2625 | 0.4124 |
| Game-theory combined weight | 0.4629 | 0.2489 | 0.2882 |
I found that surface roughness had the largest combined weight, 46.29%, followed by surface microhardness, 28.82%, and surface residual stress, 24.89%. This means that in my non-circular gear cutting evaluation, roughness remains the most important single indicator, but residual stress and microhardness are too important to ignore. A comprehensive evaluation must include all three.
Grey Comprehensive Evaluation of Gear Cutting
I used grey comprehensive evaluation to rank the sixteen gear cutting runs. I defined an ideal sequence. For surface roughness and residual stress, I chose the minimum measured value as the ideal. For microhardness, I chose the maximum measured value as the ideal. I calculated the grey relational coefficient as:
$$
\xi_{ij}=\frac{\min_i\min_j |x_{0j}-x_{ij}|+\rho\max_i\max_j |x_{0j}-x_{ij}|}{|x_{0j}-x_{ij}|+\rho\max_i\max_j |x_{0j}-x_{ij}|}
$$
I set the resolution coefficient \(\rho\) to 0.9. I then calculated the grey relational degree as:
$$
A_i=\sum_{j=1}^m w_j \xi_{ij}
$$
The grey relational coefficients and degrees are listed below.
| Run | Ra coefficient | σ coefficient | μh coefficient | Grey relational degree | Rank |
|---|---|---|---|---|---|
| 1 | 0.764 | 1.000 | 0.474 | 0.739 | 4 |
| 2 | 0.735 | 0.920 | 0.503 | 0.714 | 7 |
| 3 | 0.766 | 0.846 | 0.544 | 0.722 | 6 |
| 4 | 0.790 | 0.779 | 0.580 | 0.727 | 5 |
| 5 | 1.000 | 0.936 | 0.516 | 0.845 | 1 |
| 6 | 0.906 | 0.860 | 0.557 | 0.794 | 2 |
| 7 | 0.689 | 0.778 | 0.629 | 0.694 | 11 |
| 8 | 0.536 | 0.721 | 0.663 | 0.619 | 15 |
| 9 | 0.888 | 0.873 | 0.574 | 0.794 | 3 |
| 10 | 0.680 | 0.816 | 0.652 | 0.706 | 9 |
| 11 | 0.597 | 0.753 | 0.800 | 0.694 | 10 |
| 12 | 0.495 | 0.562 | 0.835 | 0.610 | 16 |
| 13 | 0.780 | 0.677 | 0.620 | 0.708 | 8 |
| 14 | 0.706 | 0.523 | 0.659 | 0.647 | 13 |
| 15 | 0.591 | 0.603 | 0.877 | 0.676 | 12 |
| 16 | 0.474 | 0.474 | 1.000 | 0.626 | 14 |
The maximum grey relational degree was 0.845 for run 5. This corresponded to 2 A peak current, 8 μs pulse-on time, 6 μs pulse-off time, and tracking level 4. I therefore considered this the best comprehensive surface integrity result among the sixteen gear cutting runs. I also performed variance analysis on the grey relational degree. The results showed that pulse-on time had a significant effect on the comprehensive surface integrity, while peak current, pulse-off time, and tracking were less significant. The order of influence on the comprehensive response was pulse-on time, tracking, peak current, and pulse-off time.
| Source | Type III sum of squares | Degrees of freedom | Mean square | F value | P value |
|---|---|---|---|---|---|
| Peak current | 0.013 | 3 | 0.004 | 4.333 | 0.130 |
| Pulse-on time | 0.044 | 3 | 0.015 | 14.311 | 0.028 |
| Pulse-off time | 0.004 | 3 | 0.001 | 1.244 | 0.431 |
| Tracking | 0.021 | 3 | 0.007 | 6.915 | 0.073 |
| Error | 0.003 | 3 | 0.001 | — | — |
Prediction Models for Gear Cutting Surface Integrity
After evaluating the experimental results, I built prediction models so that the surface integrity of non-circular gear cutting could be estimated without running every possible experiment. I used two approaches: multivariate nonlinear regression and a BP neural network trained with the Levenberg-Marquardt algorithm. I used peak current, pulse-on time, pulse-off time, and tracking as inputs. I used surface roughness, residual stress, and microhardness as outputs.
The multivariate nonlinear regression models included linear terms, quadratic terms, and two-factor interaction terms. The general form I used is:
$$
Y=b_0+\sum_i b_i x_i+\sum_i b_{ii}x_i^2+\sum_i\sum_{j>i} b_{ij}x_i x_j
$$
I built separate regression equations for each response. For surface roughness, I obtained:
$$
Ra=7.197+0.067x_1-0.293x_2-0.916x_3+1.655x_4
+0.004x_1x_2+0.097x_1x_3+0.184x_1x_4
+0.027x_2x_3+0.011x_2x_4+0.234x_3x_4
-0.219x_1^2-0.002x_2^2-0.019x_3^2+0.110x_4^2
$$
For residual stress, I obtained:
$$
\sigma=0.514-9.419x_1+1.971x_2+28.687x_3+16.442x_4
+0.388x_1x_2+2.197x_1x_3+4.263x_1x_4
+0.047x_2x_3+0.221x_2x_4+2.915x_3x_4
-8.719x_1^2-0.059x_2^2-1.201x_3^2+1.963x_4^2
$$
For microhardness, I obtained:
$$
\mu h=108.871+126.298x_1+11.230x_2+71.875x_3+117.445x_4
+0.407x_1x_2+0.739x_1x_3+0.974x_1x_4
+2.113x_2x_3+2.411x_2x_4+6.876x_3x_4
+27.066x_1^2+0.135x_2^2+4.096x_3^2+3.369x_4^2
$$
I evaluated the regression models by analysis of variance. The results showed very high significance and a coefficient of determination of about 0.997 for all three responses.
| Response | Regression sum of squares | Degrees of freedom | Mean square | F value | R2 |
|---|---|---|---|---|---|
| Ra | 307.438 | 15 | 20.496 | 40.992 | 0.997 |
| σ | 241097.300 | 15 | 16073.153 | 655.693 | 0.997 |
| μh | 2461697.677 | 15 | 164113.178 | 6871.260 | 0.997 |
For the BP neural network, I used a three-layer architecture with an input layer, one hidden layer, and an output layer. I used the Levenberg-Marquardt algorithm because it combines the advantages of Gauss-Newton and gradient descent. I tested hidden-layer neuron numbers from 8 to 12. I evaluated the models with root mean square error and the coefficient of determination. The best architecture was 4-10-3, which means four input neurons, ten hidden neurons, and three output neurons.
| Hidden neurons | RMSE | R2 |
|---|---|---|
| 8 | 15.9165 | 0.9978 |
| 9 | 16.8741 | 0.9970 |
| 10 | 9.5360 | 0.9979 |
| 11 | 24.4486 | 0.9970 |
| 12 | 19.0098 | 0.9970 |
The error metrics I used for the neural network were:
$$
RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^n (t_i-o_i)^2}
$$
$$
R^2=1-\frac{\sum_{i=1}^n (t_i-o_i)^2}{\sum_{i=1}^n (o_i-\bar{o})^2}
$$
I validated both prediction models with additional gear cutting tests. The additional parameters were 1 A, 8 μs pulse-on time, 8 μs pulse-off time, and tracking level 5 for the first validation, and 4 A, 32 μs pulse-on time, 8 μs pulse-off time, and tracking level 4 for the second validation. The comparison between measured and predicted values is shown below.
| Validation | Response | Measured | Neural prediction | Neural error | Regression prediction | Regression error |
|---|---|---|---|---|---|---|
| 1 | Ra | 2.175 | 2.317 | 6.53% | 1.924 | -11.54% |
| 1 | σ | 111.540 | 104.796 | -6.05% | 113.52 | 1.78% |
| 1 | μh | 347.783 | 351.697 | 1.13% | 349.445 | 0.48% |
| 2 | Ra | 5.692 | 5.509 | -3.22% | 5.341 | -6.17% |
| 2 | σ | 137.892 | 151.501 | 9.87% | 157.11 | 9.59% |
| 2 | μh | 410.930 | 429.692 | 4.57% | 378.147 | -7.98% |
I found that both models were reliable. The neural network gave better results for some responses, while the nonlinear regression gave better results for others. Together, they provide a solid basis for optimization.
Multi-Objective Optimization with the Snake Optimizer
Because the surface integrity responses conflict, I formulated a multi-objective optimization problem. I minimized surface roughness and residual stress and maximized microhardness. The optimization variables were peak current, pulse-on time, pulse-off time, and tracking. I used the regression models as objective functions. The constraints were:
$$
1 \leq x_1 \leq 5
$$
$$
5 \leq x_2 \leq 40
$$
$$
4 \leq x_3 \leq 10
$$
$$
2 \leq x_4 \leq 5
$$
$$
\mu h \geq 195
$$
The multi-objective problem was:
$$
\min f_1(x_1,x_2,x_3,x_4)=Ra(x_1,x_2,x_3,x_4)
$$
$$
\min f_2(x_1,x_2,x_3,x_4)=\sigma(x_1,x_2,x_3,x_4)
$$
$$
\max f_3(x_1,x_2,x_3,x_4)=\mu h(x_1,x_2,x_3,x_4)
$$
I used the snake optimizer because it simulates the foraging, fighting, mating, and egg-hatching behavior of snakes. It divides the population into male and female groups and switches between exploration and exploitation according to temperature and food quantity. The temperature and food quantity are:
$$
Temp=\exp(-t/T)
$$
$$
Q=c_1\exp((t-T)/T)
$$
In the exploration stage, when food is insufficient, the position update follows:
$$
X_i^{m}(t+1)=X_{rand}^{m}(t)\pm c_2 A_m (X_{\max}-X_{\min}) rand + X_{\min}
$$
$$
X_i^{f}(t+1)=X_{rand}^{f}(t)\pm c_2 A_f (X_{\max}-X_{\min}) rand + X_{\min}
$$
In the exploitation stage, when food is sufficient and temperature is high, the snake moves toward the food:
$$
X_{i,j}(t+1)=X_{food}\pm c_3 Temp rand (X_{food}-X_{i,j}(t))
$$
When temperature is low, the algorithm enters fighting or mating mode. In fighting mode, the male and female positions are updated as:
$$
X_i^{m}(t+1)=X_i^{m}(t)\pm c_3 FM rand (X_{best}^{f}-X_i^{m}(t))
$$
$$
X_i^{f}(t+1)=X_i^{f}(t)\pm c_3 FF rand (X_{best}^{m}-X_i^{f}(t))
$$
In mating mode, the positions are updated as:
$$
X_i^{m}(t+1)=X_i^{m}(t)\pm c_3 M_m rand (Q X_i^{f}(t)-X_i^{m}(t))
$$
$$
X_i^{f}(t+1)=X_i^{f}(t)\pm c_3 M_f rand (Q X_i^{m}(t)-X_i^{f}(t))
$$
I ran the optimizer with a population size of 80 and 300 iterations. The Pareto front showed that roughness, residual stress, and microhardness cannot all reach their individual best values at the same time. The Pareto set provides a practical selection space for gear cutting by WEDM. A representative subset of the Pareto-optimal parameters is shown below.
| Run | Peak current (A) | Pulse-on time (μs) | Pulse-off time (μs) | Tracking | Ra (μm) | σ (MPa) | μh (HV) |
|---|---|---|---|---|---|---|---|
| 1 | 1.7 | 5.0 | 5 | 4 | 1.94269 | 92.45018 | 336.87705 |
| 2 | 1.0 | 5.3 | 5 | 3 | 2.50932 | 92.19239 | 333.07655 |
| 3 | 1.0 | 10.9 | 5 | 2 | 4.15098 | 93.57021 | 363.97465 |
| 4 | 1.5 | 5.9 | 5 | 3 | 2.90293 | 93.08206 | 351.48575 |
| 5 | 1.0 | 7.0 | 5 | 2 | 3.57300 | 95.38800 | 354.83500 |
| 6 | 1.3 | 6.2 | 5 | 3 | 2.85165 | 92.31482 | 346.60679 |
| 7 | 1.0 | 8.1 | 5 | 3 | 2.98028 | 91.20791 | 342.53495 |
| 8 | 1.3 | 12.3 | 5 | 2 | 4.48327 | 95.64862 | 376.83661 |
| 9 | 1.0 | 15.4 | 5 | 2 | 4.74228 | 93.02256 | 372.44140 |
| 10 | 2.5 | 13.9 | 8 | 5 | 4.04613 | 96.48486 | 401.49055 |
| 11 | 2.3 | 15.2 | 8 | 5 | 4.00055 | 97.07862 | 400.86829 |
| 12 | 2.2 | 12.7 | 8 | 5 | 3.70422 | 96.69747 | 391.02001 |
| 13 | 4.4 | 5.0 | 5 | 5 | 1.25696 | 114.23892 | 322.25680 |
| 14 | 1.0 | 6.4 | 5 | 4 | 1.69748 | 93.28556 | 320.00600 |
| 15 | 1.5 | 5.0 | 5 | 3 | 2.74525 | 93.32650 | 348.26375 |
| 16 | 1.0 | 12.0 | 5 | 2 | 4.30300 | 93.28300 | 366.25000 |
| 17 | 1.5 | 6.5 | 5 | 3 | 3.00625 | 92.95600 | 353.58425 |
| 18 | 1.0 | 8.5 | 5 | 2 | 3.80250 | 94.54125 | 358.54825 |
| 19 | 1.0 | 5.0 | 5 | 4 | 1.44100 | 93.47400 | 314.02100 |
| 20 | 1.2 | 5.0 | 5 | 3 | 2.58544 | 92.51548 | 339.11120 |
Discussion of Gear Cutting Quality
My results show that WEDM is a viable method for non-circular gear cutting, but the surface integrity is strongly dependent on the discharge energy and the gap state. Peak current and pulse-on time are the primary drivers of crater size, roughness, tensile residual stress, and microhardness. Pulse-off time and tracking play secondary but important roles because they influence debris evacuation, deionization, and discharge stability. For non-circular gear cutting, the changing curvature means that these secondary effects can become locally important. A parameter set that works well on a circular gear may not produce the same surface integrity on a non-circular gear if the local gap condition changes.
I also found that surface integrity cannot be represented by roughness alone. If I minimized roughness only, I tended to reduce discharge energy, which also reduced the hardened layer and microhardness. If I maximized microhardness only, I increased discharge energy, which worsened roughness and residual stress. The combined weighting and grey evaluation gave a more balanced result. In my experiments, the best comprehensive result was obtained at 2 A peak current, 8 μs pulse-on time, 6 μs pulse-off time, and tracking level 4. This combination did not give the absolute minimum roughness or the absolute maximum hardness, but it gave the best overall surface integrity according to the combined weights.
The prediction models showed that both nonlinear regression and the BP neural network can estimate surface integrity with good accuracy. The regression model is easy to interpret, and its coefficients show the direction and magnitude of each effect. The neural network is more flexible and can capture complex nonlinear relationships, but it requires careful selection of hidden neurons to avoid underfitting or overfitting. For practical gear cutting, I would use the regression model for quick parameter screening and the neural network for more accurate prediction when enough data are available.
The snake optimizer provided a Pareto set rather than a single solution. This is useful because different non-circular gear applications have different priorities. If a gear requires low noise and good contact fatigue resistance, I would choose a solution with lower roughness and lower tensile residual stress. If a gear requires high wear resistance, I would choose a solution with higher microhardness, accepting a slight increase in roughness or residual stress. The Pareto set gives the decision maker a transparent trade-off space for gear cutting by WEDM.
Conclusions from My Study
I drew the following conclusions from my study of surface integrity in non-circular gear cutting by WEDM.
First, the single-pulse simulation showed that peak current and pulse-on time control the temperature field and crater depth. Increasing peak current increases the maximum temperature and crater depth. Increasing pulse-on time decreases the maximum temperature because the discharge channel radius increases, but it increases the crater depth. These effects explain why peak current and pulse-on time dominate the measured surface integrity in gear cutting.
Second, the orthogonal gear cutting experiments showed that surface roughness is significantly affected by peak current, pulse-on time, pulse-off time, and tracking. The order of influence is pulse-on time, peak current, tracking, and pulse-off time. Residual stress is mainly affected by peak current and pulse-on time. Microhardness is also mainly affected by peak current and pulse-on time. The optimal levels for roughness and residual stress are similar, but the optimal levels for microhardness are different. This confirms the conflict among surface integrity responses in gear cutting.
Third, the combined weighting results showed that surface roughness has the largest weight, followed by microhardness and residual stress. The grey comprehensive evaluation identified the best overall parameter combination in my experiments. Pulse-on time was the most important factor for the comprehensive surface integrity response.
Fourth, the multivariate nonlinear regression and BP neural network models both predicted surface roughness, residual stress, and microhardness with acceptable accuracy. The neural network with ten hidden neurons performed best among the tested architectures. These models provide a foundation for intelligent prediction of gear cutting quality.
Fifth, the snake optimizer produced a Pareto front for multi-objective gear cutting optimization. The Pareto set gives a practical range of parameter choices for balancing roughness, residual stress, and microhardness. This is more useful than a single optimal solution because real non-circular gear cutting requirements vary with the application.
Overall, I conclude that surface integrity in non-circular gear cutting by WEDM should be controlled as a multi-response problem. The combination of simulation, orthogonal experiments, statistical analysis, comprehensive evaluation, machine-learning prediction, and multi-objective optimization provides a systematic route for improving gear cutting quality. In future work, I would include more factors such as wire type, dielectric composition, and workpiece thickness, and I would extend the evaluation to fatigue life, load capacity, and wear performance of the non-circular gear. This would allow gear cutting by WEDM to be optimized not only for surface integrity but also for the final service performance of the gear.