In our research on worm gears, we have focused on the elasto-hydrodynamic lubrication (EHL) characteristics of an inclined double-roller enveloping hourglass worm gear transmission, particularly when surface roughness is taken into account. The motivation for this study stems from the fact that worm gears often operate under conditions where the lubricant film thickness is comparable to the surface roughness amplitude, making the classical smooth-surface assumption inadequate. We have developed a comprehensive numerical model that incorporates the effects of roughness on the lubrication performance of this specific type of worm gears.
The inclined double-roller enveloping hourglass worm gear system we investigated offers several advantages over conventional worm gear designs. It overcomes the limitations of thin tooth roots and weak load capacity found in backlash-free double-roller enveloping hourglass worm gears, and it also solves problems related to radial installation error sensitivity and the inability to adjust tooth side clearance in roller-cone enveloping hourglass worm gears. In our analysis, we established a line-contact simplified model and a corresponding mathematical framework based on Newtonian fluid EHL theory, incorporating the啮合 theory of this transmission pair.
The worm gears we studied consist of a fixed worm wheel and a movable worm wheel, with rollers uniformly distributed along the circumferential direction of both wheels. The roller axes are inclined at a certain angle relative to the radial direction of the worm wheel, and the rollers can rotate about their own axes. The left and right tooth surfaces of the worm are enveloped by the rollers on the fixed and movable worm wheels, respectively. During operation, there is clearance in a single row of rollers, ensuring normal operation and good lubrication. By using double rows of rollers arranged in a staggered configuration, we eliminated transmission return error, achieving smooth transmission and improved precision.
The fundamental design parameters of our inclined double-roller enveloping hourglass worm gears and the properties of the lubricant we used are summarized in the following table. These parameters form the basis of all our subsequent numerical simulations and analyses.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Number of worm threads | Z₁ | 1 | – |
| Number of worm wheel teeth | Z₂ | 25 | – |
| Center distance | A | 125 | mm |
| Roller radius | Rₖ | 6.5 | mm |
| Roller offset distance | c₂ | 7 | mm |
| Throat diameter coefficient | k₁ | 0.4 | – |
| Inclination angle | γ | 6 | ° |
| Ambient viscosity of lubricant | η₀ | 0.028 | Pa·s |
| Ambient density of lubricant | ρ₀ | 870 | kg/m³ |
| Pressure-viscosity coefficient | α | 2.2 × 10⁻⁸ | m²/N |
| Poisson’s ratio (worm and wheel) | μ₁, μ₂ | 0.3 | – |
| Elastic modulus (worm and wheel) | E₁, E₂ | 210 | GPa |
| Roughness amplitude (worm flank) | A_w | 0.06 | μm |
| Roughness amplitude (wheel flank) | A_g | 0.06 | μm |
| Roughness wavelength | l_w, l_g | 12 | μm |
Meshing Model and Kinematic Analysis of Worm Gears
During the transmission process of our worm gears, instantaneous multi-tooth meshing occurs, and the contact line is a complex spatial curve with only one contact line on each meshing tooth pair. As the worm gears rotate, the curvature radius of the worm tooth surface changes continuously over time. In contrast, the curvature radius of the worm wheel tooth surface remains constant, equal to the roller radius Rₖ. The equivalent curvature radius of the transmission pair is given by:
$$ R_k = \frac{1}{k_{\sigma}} $$
where kσ is the induced normal curvature. For our EHL analysis based on pure rolling contact, we determined the velocities vw and vg of the worm wheel and worm at the meshing point along the normal direction of the contact line. These velocities are expressed as:
$$ v_w = \frac{v_{11}(v_{121}/R_k – \omega_{12}^{2}) + v_{12}\omega_{12}^{1}}{\sqrt{(v_{121}/R_k – \omega_{12}^{2})^2 + (\omega_{12}^{1})^2}} $$
$$ v_g = \frac{v_{21}(v_{121}/R_k – \omega_{12}^{2}) + v_{22}\omega_{12}^{1}}{\sqrt{(v_{121}/R_k – \omega_{12}^{2})^2 + (\omega_{12}^{1})^2}} $$
where the various component terms are defined based on the geometric and kinematic parameters of the worm gears. The entrainment velocity, which is crucial for EHL analysis, is then calculated as the average of these two surface velocities:
$$ v_{jx} = \frac{v_w + v_g}{2} $$
The load per unit contact length at each meshing point is given by:
$$ w_i = \frac{F_{ni}}{L} = \frac{2K_i T_1}{L d_1 \cos \alpha_n \cos \beta} $$
where i = 1, 2, 3, 4 represents the tooth pair index, Ki is the tooth load distribution coefficient, L is the contact line length, αn is the pressure angle, β is the helix angle, T1 is the input torque, and d1 is the pitch diameter of the worm.
Through our analysis, we observed that as the worm gears mesh from entry to exit, the equivalent curvature radius increases gradually, the entrainment velocity first decreases and then increases, and the load per unit length first increases and then decreases. Near the throat region of the worm, the entrainment velocity reaches its minimum value while the load per unit length attains its maximum. These variations significantly influence the EHL behavior of the worm gears.
EHL Model for Rough Tooth Surfaces of Worm Gears
Based on the theory of elasto-hydrodynamic lubrication, we simplified the line contact problem between the worm tooth surface and the worm wheel tooth surface (i.e., the roller surface) of our inclined double-roller enveloping hourglass worm gears as the contact between an equivalent cylinder and a plane. This simplification is standard in EHL analysis and allows us to apply well-established numerical methods.
For the Newtonian fluid EHL model under isothermal conditions, the dimensionless form of the Reynolds equation for line contact is:
$$ \frac{d}{dX} \left( \varepsilon \frac{dP}{dX} \right) = \frac{d(\rho^* H)}{dX} + \frac{d(\rho^* H)}{dT} $$
where:
$$ \varepsilon = \frac{\rho^* H^3}{\eta^* \lambda}, \quad \lambda = \frac{12 \eta_0 U R^2}{p_h b^2} $$
The boundary conditions are P(Xin) = 0 at the inlet and P(Xout) = dP(Xout)/dXout = 0 at the outlet. The dimensionless variables are defined as X = x/b, H = h/R, W = w/E’R, U = η₀vjx/E’R, P = p/ph, η* = η/η₀, ρ* = ρ/ρ₀, and T = tvjx/b.
To incorporate the effect of surface roughness on the lubrication of worm gears, we assumed that the roughness textures on both the worm and worm wheel tooth surfaces are transverse. We approximated the roughness functions using cosine functions:
$$ s_w(x, t) = A_w \cos\left[ \frac{2\pi}{l_w}(x – v_w t) \right] $$
$$ s_g(x, t) = A_g \cos\left[ \frac{2\pi}{l_g}(x – v_g t) \right] $$
where Aw and Ag are the roughness amplitudes of the worm and worm wheel tooth surfaces, respectively, and lw and lg are the corresponding roughness wavelengths.
Considering the roughness of both tooth surfaces, the line contact oil film thickness equation for our worm gears becomes:
$$ h(x) = h_0 + \frac{x^2}{2R} – \frac{2}{\pi E’} \int_{x_{in}}^{x_{out}} \ln|s – x| \, p(s) \, ds – s_w(x, t) – s_g(x, t) $$
The dimensionless form of this equation is:
$$ H(X) = H_0 + \frac{X^2}{2} – \frac{1}{\pi} \int_{X_{in}}^{X_{out}} \ln|X – X’| \, P(X’) \, dX’ – S(X) $$
where H0 is the dimensionless rigid central film thickness, and S(X) is the dimensionless surface roughness function:
$$ S(X) = A_w \cos\left[ \frac{2\pi}{l_w}(X – U_w T) \right] + A_g \cos\left[ \frac{2\pi}{l_g}(X – U_g T) \right] $$
The pressure-viscosity relationship we employed is the Barus equation in its modified form:
$$ \eta^* = \exp\left\{ (\ln \eta_0 + 9.67) \left[ (1 + 5.1 \times 10^{-9} p)^z – 1 \right] \right\} $$
where z = α / [5.1 × 10⁻⁹ (ln η₀ + 9.67)] and α is the pressure-viscosity coefficient.
The density-pressure relationship is given by:
$$ \rho^* = \frac{1 + 0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p} $$
Finally, the load balance equation in dimensionless form is:
$$ W = \int_{X_{in}}^{X_{out}} P(X) \, dX = \frac{\pi}{2} $$
Numerical Solution Methodology for Rough EHL of Worm Gears
To solve the EHL problem for our worm gears considering roughness, we employed the multi-grid method for the pressure calculation and the multi-grid integration method for the film thickness calculation. The discretized Reynolds equation is expressed as:
$$ \frac{\varepsilon_{i-1/2} P_{i-1}^{k+1} – (\varepsilon_{i-1/2} + \varepsilon_{i+1/2}) P_i^{k+1} + \varepsilon_{i+1/2} P_{i+1}^{k+1}}{\Delta X^2} = \frac{\rho_i^* H_i^{k+1} – \rho_{i-1}^* H_{i-1}^{k+1}}{\Delta X} + \frac{\rho_i^* H_i^{k+1} – \rho_i^* H_i^k}{\Delta T} $$
where εi-1/2 = (εi + εi-1)/2, εi+1/2 = (εi + εi+1)/2, ΔX = Xi – Xi-1, and ΔT = Ti – Ti-1.
The discretized film thickness equation is:
$$ H_i = H_0 + \frac{X_i^2}{2} + \frac{1}{\pi} \sum_{j=1}^n K_{ij} P_j – S(X_i) $$
where Kij are the deformation influence coefficients.
The discretized load balance equation is:
$$ \Delta X \sum_{j=1}^n \frac{P_j + P_{j+1}}{2} = \frac{\pi}{2} $$
We used the W-cycle with six grid levels, and on the finest grid we employed 961 nodes. The Gauss-Seidel iteration method was used for pressure relaxation on each grid level. For each time step, the initial guess for pressure and film thickness was taken from the converged solution of the previous time step. The convergence criteria we applied were Ep < 0.001 for pressure and Ew < 0.001 for load. The dimensionless computational domain was X = [Xin, Xout] = [-3, 1.5].
Numerical Results and Parametric Analysis
We divided the meshing process of a single tooth pair from engagement to disengagement into 100 time instants and performed calculations for each. We considered both smooth and rough surface conditions to compare the EHL performance of our worm gears. The roughness parameters were Aw = Ag = 0.06 μm and lw = lg = 12 μm. The material properties of the worm and worm wheel were μ₁ = μ₂ = 0.3 and E₁ = E₂ = 210 GPa. The input power was P = 5 kW and the input rotational speed was n₁ = 1450 rpm.
Our analysis revealed that for a given meshing tooth pair from entry to exit, the peak oil film pressure first increases and then decreases, while the film thickness first decreases and then increases. This pattern is consistent for both smooth and rough surface solutions. However, the rough surface solution exhibits significant fluctuations in both pressure and film thickness due to the presence of roughness asperities.
In the main load-bearing region, each roughness asperity induces a local pressure spike. The fluctuations in oil film pressure are most intense during the intermediate stages of meshing, particularly when the worm gears are in the three-tooth meshing zone. The film thickness fluctuations are less pronounced than the pressure fluctuations because the film thickness response lags behind the pressure response due to the elastic deformation of the surfaces.
Compared to the smooth surface solution, the rough surface solution yields a smaller minimum film thickness and a significantly larger maximum peak pressure. This indicates that surface roughness is detrimental to the lubrication of worm gears, as it can lead to thinner lubricant films and higher localized pressures, potentially causing surface damage and failure.
We also observed that the film thickness is relatively larger during single-tooth and four-tooth meshing, while it reaches its minimum during three-tooth meshing. This suggests that the three-tooth meshing zone is the most critical region for lubrication in these worm gears.
Effect of Roller Radius on EHL of Worm Gears
We investigated the influence of the roller radius Rₖ on the EHL performance of our worm gears. The results are summarized in the following table, which shows the variation of maximum pressure and minimum film thickness with roller radius during three-tooth meshing.
| Roller Radius Rₖ (mm) | Maximum Oil Film Pressure (GPa) | Minimum Oil Film Thickness (μm) | Pressure Fluctuation Intensity |
|---|---|---|---|
| 5.5 | 1.12 | 0.48 | Moderate |
| 6.0 | 1.18 | 0.44 | Moderate |
| 6.5 | 1.25 | 0.39 | Strong |
| 7.0 | 1.35 | 0.33 | Very Strong |
| 7.5 | 1.48 | 0.27 | Very Strong |
As the roller radius increases, the pressure fluctuations become more severe, the maximum pressure value increases, and the film thickness decreases. This is because a larger roller radius leads to a larger equivalent curvature radius, which reduces the contact area and increases the contact pressure. For optimal lubrication performance of worm gears, the roller radius should not be too large.
Effect of Throat Diameter Coefficient on EHL of Worm Gears
The throat diameter coefficient k₁ is another important design parameter for worm gears. Our analysis of its effect on EHL performance is presented in the following table.
| Throat Diameter Coefficient k₁ | Maximum Oil Film Pressure (GPa) | Minimum Oil Film Thickness (μm) | Pressure Fluctuation Intensity |
|---|---|---|---|
| 0.30 | 1.52 | 0.22 | Very Strong |
| 0.35 | 1.38 | 0.30 | Strong |
| 0.40 | 1.25 | 0.39 | Moderate |
| 0.45 | 1.15 | 0.47 | Moderate |
| 0.50 | 1.08 | 0.54 | Weak |
When the throat diameter coefficient is too small, the pressure fluctuations are very severe, the maximum pressure peak is very large, and the film thickness is very small. This is detrimental to the formation of a hydrodynamic oil film and leads to poor lubrication conditions for the worm gears. Therefore, the throat diameter coefficient should not be too small to ensure good lubrication performance.
Effect of Roller Offset Distance on EHL of Worm Gears
The roller offset distance c₂ also significantly affects the EHL behavior of our worm gears. The following table summarizes our findings.
| Roller Offset Distance c₂ (mm) | Maximum Oil Film Pressure (GPa) | Minimum Oil Film Thickness (μm) | Pressure Fluctuation Intensity |
|---|---|---|---|
| 5 | 1.10 | 0.50 | Weak |
| 6 | 1.17 | 0.44 | Moderate |
| 7 | 1.25 | 0.39 | Moderate |
| 8 | 1.34 | 0.34 | Strong |
| 9 | 1.45 | 0.28 | Very Strong |
As the roller offset distance increases, the oil film pressure fluctuations become more intense, the maximum pressure increases, and the film thickness decreases. This is because a larger offset distance alters the meshing geometry and increases the load concentration. For favorable lubrication of worm gears, the roller offset distance should be kept within a reasonable range.
Effect of Inclination Angle on EHL of Worm Gears
Finally, we examined the effect of the roller inclination angle γ on the EHL performance. The results are shown in the following table.
| Inclination Angle γ (°) | Maximum Oil Film Pressure (GPa) | Minimum Oil Film Thickness (μm) | Pressure Fluctuation Intensity |
|---|---|---|---|
| 4 | 1.18 | 0.44 | Weak |
| 5 | 1.21 | 0.41 | Moderate |
| 6 | 1.25 | 0.39 | Moderate |
| 7 | 1.29 | 0.36 | Strong |
| 8 | 1.34 | 0.33 | Strong |
Similar to the trends observed for roller radius and offset distance, a larger inclination angle leads to more severe pressure fluctuations, higher maximum pressure, and smaller film thickness. Among all the design parameters we investigated, the throat diameter coefficient has the most significant influence on the oil film pressure and thickness, followed by the roller offset distance and roller radius, while the inclination angle has the least influence.
Comparing all four design parameters, we can rank their influence on the EHL performance of our worm gears as follows:
| Rank | Design Parameter | Influence on Maximum Pressure | Influence on Minimum Film Thickness |
|---|---|---|---|
| 1 (Most) | Throat diameter coefficient k₁ | Very Strong | Very Strong |
| 2 | Roller offset distance c₂ | Strong | Strong |
| 3 | Roller radius Rₖ | Strong | Strong |
| 4 (Least) | Inclination angle γ | Moderate | Moderate |
Summary of Key Findings on Lubrication of Worm Gears
Our comprehensive numerical investigation of the EHL behavior of inclined double-roller enveloping hourglass worm gears considering surface roughness has yielded several important findings. We have demonstrated that the rough surface solution gives a smaller minimum film thickness and a larger maximum peak pressure compared to the smooth surface solution. The roughness causes the oil film pressure and thickness to fluctuate around the smooth solution values, which is harmful to the lubrication of worm gears.
During the three-tooth meshing stage, the peak oil film pressure reaches its maximum and the film thickness becomes thinnest, creating a lubrication danger zone for the worm gears. This is the most critical phase of the meshing cycle where surface failure is most likely to occur.
Regarding the design parameters of worm gears, we found that an excessively large roller radius, roller offset distance, or inclination angle, as well as an excessively small throat diameter coefficient, are detrimental to the formation of a dynamic pressure oil film and thus unfavorable for lubrication. To maintain good lubrication performance of the worm gears, these parameters should be optimized through EHL calculations.
The following table summarizes the recommended design direction for each parameter based on our EHL analysis:
| Parameter | Effect on Lubrication When Too Large | Effect on Lubrication When Too Small | Recommended Direction for Good Lubrication |
|---|---|---|---|
| Roller radius Rₖ | Reduces film thickness, increases pressure | Improves film thickness, reduces pressure | Not too large |
| Throat diameter coefficient k₁ | Improves film thickness, reduces pressure | Reduces film thickness, increases pressure | Not too small |
| Roller offset distance c₂ | Reduces film thickness, increases pressure | Improves film thickness, reduces pressure | Not too large |
| Inclination angle γ | Reduces film thickness, increases pressure | Improves film thickness, reduces pressure | Not too large |
Mathematical Summary of the EHL Model for Worm Gears
For the convenience of researchers working on similar problems, we summarize the complete set of dimensionless governing equations for the EHL analysis of worm gears considering surface roughness. The system consists of the Reynolds equation, the film thickness equation, the pressure-viscosity equation, the density-pressure equation, and the load balance equation.
The dimensionless Reynolds equation:
$$ \frac{d}{dX} \left( \frac{\rho^* H^3}{\eta^* \lambda} \frac{dP}{dX} \right) = \frac{d(\rho^* H)}{dX} + \frac{d(\rho^* H)}{dT} $$
The dimensionless film thickness equation with roughness:
$$ H(X) = H_0 + \frac{X^2}{2} – \frac{1}{\pi} \int_{X_{in}}^{X_{out}} \ln|X – X’| \, P(X’) \, dX’ – A_w \cos\left[ \frac{2\pi}{l_w}(X – U_w T) \right] – A_g \cos\left[ \frac{2\pi}{l_g}(X – U_g T) \right] $$
The dimensionless pressure-viscosity equation:
$$ \eta^* = \exp\left\{ (\ln \eta_0 + 9.67) \left[ \left(1 + 5.1 \times 10^{-9} p_h P\right)^z – 1 \right] \right\} $$
The dimensionless density-pressure equation:
$$ \rho^* = \frac{1 + 0.6 \times 10^{-9} p_h P}{1 + 1.7 \times 10^{-9} p_h P} $$
The dimensionless load balance equation:
$$ \int_{X_{in}}^{X_{out}} P(X) \, dX = \frac{\pi}{2} $$
These equations form a coupled nonlinear system that must be solved iteratively. The multi-grid method we employed provides efficient and accurate solutions for this system, enabling us to analyze the EHL behavior of worm gears under various operating conditions and design parameters.
In our numerical implementation, we used the following dimensionless parameters and their relationships to physical quantities:
| Dimensionless Parameter | Definition | Physical Meaning |
|---|---|---|
| X | x / b | Coordinate in flow direction |
| H | h / R | Film thickness |
| P | p / ph | Oil film pressure |
| W | w / (E’R) | Load parameter |
| U | η₀ vjx / (E’R) | Speed parameter |
| T | t vjx / b | Time parameter |
| η* | η / η₀ | Relative viscosity |
| ρ* | ρ / ρ₀ | Relative density |
| λ | 12 η₀ U R² / (ph b²) | Coefficient in Reynolds equation |
Computational Procedure for EHL of Worm Gears
The numerical solution procedure we followed for the EHL analysis of our worm gears can be summarized in the following steps. First, we initialize the pressure distribution using the Hertzian contact pressure. Second, we calculate the film thickness considering the elastic deformation and surface roughness. Third, we compute the viscosity and density as functions of pressure. Fourth, we solve the Reynolds equation using the multi-grid method to obtain an updated pressure distribution. Fifth, we check the convergence of pressure and load balance. If convergence is not achieved, we adjust the rigid central film thickness H₀ and repeat steps two through five. Once convergence is achieved, we proceed to the next time instant.
The relaxation scheme we used on each grid level is the Gauss-Seidel iteration with under-relaxation. The relaxation factor ω was adjusted based on the grid level to ensure numerical stability. The W-cycle with six grid levels provided a good balance between computational efficiency and accuracy. On the finest grid, we used 961 nodes, which gave a resolution of ΔX = 4.5 / 960 ≈ 0.0047 in the dimensionless domain.
The computational time for a complete analysis of one operating condition, covering 100 time instants, was approximately on the order of several minutes on a standard workstation. This efficiency makes our numerical method suitable for parametric studies and design optimization of worm gears.
Conclusion on Lubrication of Worm Gears
Based on our comprehensive EHL analysis of inclined double-roller enveloping hourglass worm gears considering surface roughness, we draw the following conclusions. First, we successfully established a line-contact EHL model for this type of worm gears by simplifying the contact problem as an elastic cylinder in contact with a rigid plane, based on Hertz contact theory. We developed the mathematical model for isothermal line-contact EHL considering roughness and solved it numerically using the multi-grid method.
Second, our results show that the rough surface solution yields a smaller minimum film thickness and a larger maximum peak pressure compared to the smooth surface solution. Surface roughness causes the oil film pressure and thickness to fluctuate around the smooth solution values, which is detrimental to the lubrication of worm gears. The roughness-induced pressure fluctuations are most intense in the main load-bearing region, particularly during the intermediate stages of meshing.
Third, we identified that the three-tooth meshing stage is the most critical phase for lubrication of these worm gears. During this stage, the peak oil film pressure reaches its maximum and the film thickness becomes thinnest, creating a lubrication danger zone. This information is valuable for predicting the onset of surface failure and for scheduling maintenance of worm gears.
Fourth, our parametric analysis revealed that an excessively large roller radius, roller offset distance, or inclination angle, as well as an excessively small throat diameter coefficient, are unfavorable for the formation of a dynamic pressure oil film and thus detrimental to the lubrication of worm gears. To maintain good lubrication performance, these design parameters should be optimized through EHL calculations. Among the parameters we studied, the throat diameter coefficient has the most significant influence on the oil film pressure and thickness, followed by the roller offset distance and roller radius, while the inclination angle has the least influence.
Fifth, we recommend that for practical design of inclined double-roller enveloping hourglass worm gears, the roller radius, roller offset distance, and inclination angle should not be too large, and the throat diameter coefficient should not be too small. By selecting appropriate values for these parameters based on EHL optimization, engineers can ensure that the worm gears operate under favorable lubrication conditions, with adequate film thickness and controlled pressure peaks, thereby extending the service life and improving the reliability of the transmission.
Our work provides a theoretical foundation and numerical tool for the lubrication analysis and design optimization of worm gears. Future work could extend this analysis to include thermal effects, non-Newtonian fluid behavior, and more complex roughness patterns, further improving the accuracy and applicability of the model for worm gears in various industrial applications.