In this research, I focused on enhancing the lubrication and friction performance of spur gears by introducing an elliptical surface texture on the tooth flanks. Spur gears are widely used in precision transmission systems due to their reliable operation and high accuracy, yet their performance is often limited by insufficient lubrication and severe friction wear during meshing. To address these issues, I proposed an elliptical texture design and systematically studied its effects on both hydrodynamic lubrication and friction behavior. The work involved constructing a line-contact model for the equivalent contact of spur gears, solving the Reynolds equation via the MATLAB PDE toolbox, conducting friction and wear experiments on cylindrical rollers with elliptical textures, and finally verifying the findings with actual spur gear tests. The results demonstrate that a well-designed elliptical texture can significantly improve oil film pressure distribution, increase load-carrying capacity, and reduce friction and wear.
My investigation began with establishing a theoretical foundation. Since the meshing of spur gears is complex, I simplified the gear contact into an equivalent line-contact model using two cylinders of different radii, representing the instantaneous contact condition at a certain meshing point. The radii were chosen as 20 mm and 15 mm for the large and small rollers respectively, which is consistent with the MMS-2A friction and wear tester specifications. The equivalent radius of curvature was calculated to be 8.5714 mm. This line-contact model allowed me to simulate the hydrodynamic lubrication state of the gear tooth surface without dealing with the intricate involute geometry.
For the lubrication analysis, I applied the Reynolds equation, which governs the pressure distribution in a thin lubricating film. In a steady-state, isothermal condition, the equation is expressed as:
$$ \frac{\partial}{\partial x}\left(\frac{\rho h^3}{\eta} \frac{\partial p}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{\rho h^3}{\eta} \frac{\partial p}{\partial y}\right) = 6u \frac{\partial (\rho h)}{\partial x} $$
where \(p\) is the oil film pressure, \(h\) is the film thickness, \(\eta\) is the dynamic viscosity, \(\rho\) is the density, and \(u\) is the entrainment velocity. I used dimensional analysis to non-dimensionalize the equation. By defining dimensionless variables \(X=x/Ra\), \(Y=y/Ra\), \(H=h/h_0\), \(P=p/p_0\) and a parameter \(M=6\eta u Ra / (h_0^2 p_0)\), the Reynolds equation becomes:
$$ \frac{\partial}{\partial X}\left(H^3 \frac{\partial P}{\partial X}\right) + \frac{\partial}{\partial Y}\left(H^3 \frac{\partial P}{\partial Y}\right) = M \frac{\partial H}{\partial X} $$
I then transformed this equation into a standard elliptic PDE form suitable for the MATLAB PDE toolbox. The corresponding elliptic equation is:
$$ -\nabla \cdot (c \nabla z) + a z = f $$
with coefficients \(c = H^3\), \(a = 0\), and \(f = -M \partial H / \partial X\). I applied Dirichlet boundary conditions where the pressure at the edges equals the ambient pressure. The computational domain was discretized using a triangular mesh, and a grid independence study was performed to ensure accurate results. The mesh size was refined until the relative change in maximum pressure was less than 0.1%; a mesh size of 0.15625 μm was ultimately selected.
The elliptical texture geometry was modeled as a single period within a rectangular unit cell. The texture parameters included the major radius \(Ra\), the minor radius \(Rb\), the depth \(H\), the orientation angle \(\theta\), and the area density \(\delta\). I employed an arc-shaped cross-section for the elliptical dimple to better represent the actual laser-ablated profile. The film thickness within the textured region was described by an additional term that depends on the local coordinates. The area density \(\delta\) was defined as the ratio of the textured area to the unit cell area. The ellipticity ratio \(\gamma = Rb/Ra\) was also introduced. These parameters were systematically varied in order to identify their influence on the lubrication performance.
To evaluate the lubrication capability, I defined several output metrics. The dimensionless oil film pressure \(p^*\) was obtained from the PDE solution. The oil film load capacity \(W^*\) was calculated by integrating the positive pressure over the cell area:
$$ W^* = \frac{1}{T^2} \int_{A} p^* \, dA $$
Similarly, the wall shear force \(F^*\) was obtained by integrating the shear stress over the wall
$$ F^* = \frac{1}{T^2} \int_{A} \tau^* \, dA $$
Finally, a hydrodynamic lubrication performance parameter \(K = W^* / F^*\) was used to measure the overall lubrication quality. A higher \(K\) value indicates a better ability to generate a load-supporting pressure while maintaining low frictional shear.
My simulations first compared the pressure distribution of the smooth (untextured) model and the elliptical textured model. The smooth model exhibited a typical converging-diverging pressure profile, with the maximum pressure occurring near the inlet and outlet of the contact zone. In contrast, the elliptical texture introduced an additional convergence-divergence pattern inside the dimple. The pressure decreased before the dimple, reached a negative minimum at the inlet edge, then increased gradually through the dimple, and peaked at the outlet edge before decreasing again. Thus, the overall pressure distribution became a sequence of convergence – divergence – convergence – divergence. This behavior is beneficial for generating additional hydrodynamic pressure due to the wedge effect created by the textured geometry.
I also compared the lubrication performance of elliptical textures with that of circular dimples and grooves, while keeping the same area density, depth, and other parameters. The results showed that the elliptical texture provided the highest load capacity and the best \(K\) value among the three texture types. The elliptical shape combines the advantages of isolated dimples, which easily generate hydrodynamic pressure, and continuous grooves, which have a strong directional flow effect. This combination makes the elliptical texture particularly effective for improving the lubrication of spur gears.
Then I proceeded to investigate the effect of each geometric parameter. The parametric study covered the dimensionless area density \(\delta^*\), ellipticity \(\gamma\), orientation angle \(\theta\), dimensionless entrainment velocity \(U^*\), dimensionless major radius \(Ra^*\), and dimensionless texture depth \(H^*\). The selected ranges for these parameters were based on practical manufacturing limits and previous literature.
First, the effect of area density was examined. The dimensionless area density \(\delta^*\) was varied from 1 to 3 (i.e., 10% to 30%). The results, shown in Figure 3-5 (omitted here), indicated that both the oil film load capacity \(W^*\) and the performance parameter \(K\) initially decreased and then increased as \(\delta^*\) increased. The wall shear force \(F^*\) remained relatively stable. The maximum \(K\) value was achieved at \(\delta^* = 3\). I found that when the area density is too small, the texture is too sparse to produce a noticeable effect; when \(\delta^*\) lies between 2 and 3, the lubrication performance improves significantly. However, since a high area density reduces the total contact area and thereby increases contact stress, the recommended optimal range was \(\delta^* = 0.25–0.3\), i.e., 25% to 30%, as used in later experiments.
Next, I studied the influence of the ellipticity ratio \(\gamma\). I varied \(\gamma\) from 0.5 to 0.9 while keeping other parameters fixed. The results showed that the oil film load capacity \(W^*\) increased with \(\gamma\), but the wall shear force \(F^*\) and the lubrication parameter \(K\) changed only mildly. The highest \(K\) was observed at \(\gamma = 0.8\). A smaller \(\gamma\) strengthens the directionality of the ellipse but also shortens the minor radius, which reduces the pressure build-up area. Therefore, I determined that values of \(\gamma\) between 0.7 and 0.9 offer the best compromise.
The orientation angle \(\theta\) was varied from 0° to 90° in steps of 15°. For angles between 0° and 45°, both \(W^*\) and \(F^*\) fluctuated only mildly, and the performance parameter \(K\) increased steadily to a maximum at \(\theta = 45^\circ\). Beyond 45°, the load capacity dropped sharply while the shear force increased, leading to a significant reduction in \(K\). Hence, an orientation angle between 0° and 45° is recommended to maintain robust lubrication performance.
Another important factor is the entrainment velocity \(U\). I tested five values: \(U^* = 1\), 1.73, 2.25, 3.455, and 5. These corresponded to rotational speeds from 100 to 500 rpm in the roller experiments. The results showed that both \(W^*\) and \(F^*\) increased monotonically with \(U^*\), as expected, because higher velocities enhance both the shear and pressure flows. However, the ratio \(K\) changed very little, indicating that the entrainment speed has only a minor influence on the overall lubrication performance. Nevertheless, for the experimental conditions used in the roller tests, I chose \(U^* = 3.455\) (i.e., 400 rpm for the large roller) as the reference speed.
Regarding the major radius \(Ra^*\), I varied it from 1 to 8 (i.e., 40 to 320 μm). The load capacity \(W^*\) increased with \(Ra^*\) up to \(Ra^* = 6\), after which the growth slowed down. The wall shear force \(F^*\) showed a fluctuating pattern. The lubrication parameter \(K\) increased sharply for \(Ra^* < 6\) and then became almost flat or slightly decreased for larger radii. This suggests that an overly large texture can cause oil to escape from the contact region, and there is also a limit to how much of the texture can be covered by the small contact area in actual gear meshing. I thus recommended \(Ra^* = 6–8\), which corresponds to a major radius of about 120–160 μm.
The texture depth \(H^*\) was varied from 1 to 20. The results showed that both \(W^*\) and \(K\) decreased monotonically with increasing depth. The decrease was small for \(Ra^* = 1\) and 2, but was pronounced for \(Ra^* = 4\), 6, and 8 when \(H^*\) was small (less than 5). This indicates that the depth is particularly critical for larger texture sizes. A shallow texture with \(H^* = 1\) (i.e., depth equal to the minimum film thickness, 1 μm) gave the best lubrication performance. However, in practice, a very shallow texture may not provide sufficient oil reservoir capacity for wear particle entrapment. Therefore, after considering both the lubrication calculations and the experimental fabrication limitations, I selected a depth of 20 μm for the gear tests, based on the combined outcome of the friction experiments.
Table 3-1 summarizes the recommended optimal dimensionless ranges for each parameter from the lubrication analysis:
| Parameter | Recommended range |
|---|---|
| Area density \(\delta^*\) | 0.25 – 0.3 |
| Ellipticity \(\gamma\) | 0.7 – 0.9 |
| Orientation angle \(\theta\) | 0° – 45° |
| Entrainment velocity \(U^*\) | 3.455 |
| Major radius \(Ra^*\) | 6 – 8 |
| Texture depth \(H^*\) | 1 (for lubrication) |
After completing the theoretical analysis, I proceeded to experimental validation using cylindrical roller specimens. The rollers were made of 40Cr alloy steel, hardened to 33–37 HRC, with diameters of 30 mm and 40 mm for the small and large rollers respectively. The small roller was textured with elliptical patterns using a picosecond laser system. The laser parameters were optimized to achieve a good surface quality: a frequency of 400 kHz, a scanning speed of 150 mm/s, a single scan, and a unidirectional fill with a line width of 0.001 mm. The actual texture geometries were inspected with a Zygo New View 9000 white-light interferometer. The machined shapes closely matched the designed dimensions, with a maximum deviation of about 10 μm for depth and 3 μm for the major radius.
I designed an orthogonal experiment with four factors at three levels. The factors were the texture depth \(H\) (5, 10, 20 μm), the major radius \(Ra\) (40, 80, 150 μm), the ellipticity \(\gamma\) (0.5, 0.7, 0.9), and the orientation angle \(\theta\) (0°, 45°, 90°). The area density was fixed at 25% and the entrainment speed corresponded to 400 rpm for the large roller. Nine groups of textured rollers were prepared, plus one untextured control group. Each test was repeated three times, giving 30 tests in total. The friction and wear tests were carried out on an MMS-2A test rig under oil drip lubrication with L-CKC 68 gear oil. The applied load was 1600 N, the large roller speed was 400 rpm, and the small roller speed was 90% of that, resulting in a slide-roll ratio of 18.2%. Each test lasted 30 minutes. The friction coefficient was recorded continuously, and the mass loss of the small roller was measured using an analytical balance before and after each test.
The wear mass loss results are presented in Table 4-3. The highest mass loss occurred in group 4 (13.12 mg), which had the combination of \(H=10\,\mu m\), \(Ra=40\,\mu m\), \(\gamma=0.9\), \(\theta=90^\circ\). The lowest mass loss was observed in group 2 (2.61 mg) with \(H=5\,\mu m\), \(Ra=80\,\mu m\), \(\gamma=0.7\), \(\theta=90^\circ\). The untextured roller exhibited a mass loss of about 8.4 mg. The range analysis showed that the order of influence on wear loss was \(H > Ra > \gamma > \theta\). The optimal level for each factor was \(H = 5\,\mu m\), \(Ra = 150\,\mu m\), \(\gamma = 0.7\), and \(\theta = 0^\circ\), resulting in the best anti-wear combination \(H_1 Ra_3 \gamma_2 \theta_1\).
To further understand the wear mechanisms, I examined the worn surfaces using a scanning electron microscope (SEM). The observed wear forms were mainly pitting and slight scoring. Groups 2, 5, 6, and 7 showed relatively mild surface damage with only a few pits. Groups 1, 3, and 9 had more pitting, while groups 4, 8, and the untextured roller exhibited extensive and larger pits. The better wear resistance of the shallow textured rollers can be attributed to the ability of the texture to store wear debris and reduce the damaging effect of third-body abrasion. A proper texture depth and size can also improve the local oil film, reducing the severity of asperity contact.
Regarding the friction coefficient, I analyzed the average friction coefficient during the steady-state period (from 2 to 28 minutes). The untextured roller had an average friction coefficient of approximately 0.0068 in this test, but the actual values varied slightly across the repeated runs. The best friction performance was found for group 9 (\(H=20\,\mu m\), \(Ra=150\,\mu m\), \(\gamma=0.5\), \(\theta=90^\circ\)) with an average friction coefficient of 0.01045, which was 38.7% lower than the untextured control. Groups 2 and 7 also showed reductions of about 35.6% and 35.9%, respectively. The range analysis for the average friction coefficient gave the order \(H > \gamma > Ra > \theta\), with the optimal levels \(H = 20\,\mu m\), \(Ra = 80\,\mu m\), \(\gamma = 0.7\), and \(\theta = 45^\circ\). The corresponding best combination was \(H_3 Ra_2 \gamma_2 \theta_2\).
To summarize the experimental findings, I prepared Table 4-5 that lists the optimal parameter combinations for both wear and friction:
| Objective | Optimal combination | Analytical result |
|---|---|---|
| Minimize wear loss | H=5 μm, Ra=150 μm, γ=0.7, θ=0° | Lowest wear and pitting |
| Minimize friction coefficient | H=20 μm, Ra=80 μm, γ=0.7, θ=45° | Lowest average friction coefficient |
| Compromise | H=20 μm, Ra=150 μm, γ=0.7, θ=0° | Balanced wear and friction performance |
I then integrated the lubrication simulation results with the roller test results to select a final set of texture parameters for the gear experiments. The lubrication study suggested a small depth for the best hydrodynamic effect, but the wear experiments indicated that a moderate depth (20 μm) provides a better balance between wear resistance and friction reduction, likely because a deeper texture can accommodate more wear particles and provide a continuous oil supply. The major radius of 150 μm was chosen because it gave the lowest wear loss and still maintained good lubrication. The ellipticity of 0.7 and an orientation angle of 0° (aligned with the sliding direction) proved to be the most robust choices in both the simulations and the experiments. Thus, the final optimal parameters were \(H = 20\,\mu m\), \(Ra = 150\,\mu m\), \(\gamma = 0.7\), and \(\theta = 0^\circ\).
The final stage of my research involved applying this unique texture to actual spur gears. The test gears were made of 20CrMo steel, with a module of 4.5, a pressure angle of 20°, 16 teeth for the pinion and 24 teeth for the gear, a face width of 20 mm, and a center distance of 91.5 mm. The pinion was designed with a profile shift coefficient of +0.8532, and the gear with −0.5. The gears were case-hardened to 60–62 HRC on the surface and 33–37 HRC at the core. The elliptical texture was laser-machined on both flanks of all 16 pinion teeth, using the selected parameters. A texture-free pinion was used as a control.
The gear friction and wear experiments were conducted on an MFZG-1W gear testing machine, which is a closed-power-loop system. The pinion speed was set to 1400 rpm. The lubricant was the same L-CKC 68 oil, initially at 30 °C. Before the actual test, a 30-minute no-load running-in period was performed to allow the tooth surfaces to settle. Then a stepwise load test was implemented according to the Chinese standard GB/Z 13672-2022, covering 12 load levels from 9.5 N·m to 350.6 N·m. Each load level was maintained for 15 minutes. After the entire test, the gears were disassembled, cleaned ultrasonically, and examined with a VHX-2000 digital microscope at 100× magnification. The observed area was focused on the region from the pitch line to the tooth tip, where scuffing is most likely to initiate.
The surface damage morphology of the untextured gear showed wide linear or banded scratches, indicating that scuffing had occurred. In contrast, the elliptical textured gear exhibited only small point-like or narrow banded damage, and no severe scuffing was observed. The damage area ratio was calculated as the percentage of the damaged area within a fixed observed area of 9.0281 mm². For the untextured gear, the maximum damage area ratio was 8.527% (tooth #8) and the minimum was 2.509% (tooth #7). For the elliptical textured gear, the maximum was 1.611% (tooth #3) and the minimum was 0.603% (tooth #4). This corresponds to reductions of 81.11% and 76.97%, respectively. The damage area ratios across all 16 teeth of the textured gear remained below 2%, while those of the untextured gear varied from about 2.5% to 8.5%, showing a much larger scatter. These results confirm that the elliptical texture significantly improves the scuffing resistance and friction performance of spur gears.
In conclusion, my research demonstrated that elliptical surface texturing is a highly effective method for enhancing the hydrodynamic lubrication and reducing friction and wear of spur gears. By combining theoretical lubrication calculations, roller-on-disk friction tests, and gear scuffing tests, I was able to identify a set of optimized texture parameters that provide substantial benefits. The final texture parameters—depth \(20\,\mu m\), major radius \(150\,\mu m\), ellipticity \(0.7\), and orientation angle \(0^\circ\)—achieved a reduction of about 77% to 81% in the damaged area of the gear tooth surfaces compared to untextured gears. These findings offer valuable guidance for the design and application of surface texture in spur gears, and they also provide a foundation for future work on other texture shapes and more complex gear geometries.