The pursuit of high-performance, reliable, and compact speed reducers for industrial robot joints is a continuous endeavor in mechanical engineering. While RV and harmonic drives dominate the current landscape, their complexity and reliance on imported technology have spurred research into alternative compact, high-load-capacity solutions. One promising candidate is the family of planar gear enveloping worm gear drives. In these transmissions, the worm wheel tooth surface is a simple plane, which acts as the tool to generate the conjugated worm thread. This geometry offers advantages such as a high number of instantaneous contact lines, superior load-carrying capacity, and high transmission efficiency, making it suitable for robotic applications.
This article focuses on a specific variant known as the planar internal gear enveloping crown worm gear drive. Extensive prior research has established its meshing theory, manufacturing techniques, error measurement, and prototype testing, providing a solid foundation for this study. However, a detailed investigation into its lubrication performance under heavy load and high-speed conditions is crucial for ensuring reliability and longevity. The objective of this analysis is to master the thermal elastohydrodynamic lubrication (TEHL) characteristics of the conjugate tooth pairs during meshing. By integrating meshing theory with EHL theory and employing advanced numerical techniques, this work solves the line-contact TEHL problem for this transmission. The results provide detailed performance parameters—oil film thickness, pressure, and temperature rise—and systematically investigate the influence of key design parameters, namely the base circle radius, normal module, and generating plane inclination angle, on these TEHL characteristics.
1. Meshing Model of the Transmission Pair
The geometric configuration of the planar internal gear enveloping crown worm gear drive is defined by specific coordinate systems and parameters. A fixed coordinate system \( S_{1′}(O_{1′};\mathbf{i}_{1′},\mathbf{j}_{1′},\mathbf{k}_{1′}) \) is attached to the worm, and another, \( S_{2′}(O_{2′};\mathbf{i}_{2′},\mathbf{j}_{2′},\mathbf{k}_{2′}) \), is attached to the internal gear. Moving coordinate systems \( S_1(O_1;\mathbf{i}_1,\mathbf{j}_1,\mathbf{k}_1) \) and \( S_2(O_2;\mathbf{i}_2,\mathbf{j}_2,\mathbf{k}_2) \) rotate with the worm and gear, respectively, with angular velocities \( \omega_1 \) and \( \omega_2 \). A local coordinate system \( S_p(O_p;\mathbf{e}_1,\mathbf{e}_2,\mathbf{n}) \) is established at the potential contact point \( O_p \) on the base circle. Here, \( \mathbf{e}_1 \) and \( \mathbf{e}_2 \) lie on the generating plane (the planar gear tooth surface), and \( \mathbf{n} \) is the unit normal. The key parameters are: the center distance \( A \), the base circle radius \( r_b \), the generating plane inclination angle \( \beta \), the number of worm threads \( Z_1 \), the number of gear teeth \( Z_2 \), and the transmission ratio \( i_{21} = \omega_2 / \omega_1 = Z_1 / Z_2 \). The angular displacements are \( \varphi_1 \) and \( \varphi_2 \).
The equation of the planar internal gear surface (the generating plane) in \( S_p \) is given by:
$$ \mathbf{r}(u,v) = x_p\mathbf{e}_1 + y_p\mathbf{e}_2 + z_p\mathbf{n} $$
with coordinates:
$$ x_p = u, \quad y_p = v, \quad z_p = 0 $$
where \( u \) and \( v \) are surface parameters.
According to gearing theory, the meshing function \( \Phi(u, v, \varphi_2) \) for this worm gear drive is derived as:
$$ \Phi(u,v,\varphi_2) = u \sin\beta \cos\varphi_2 + (r_b \sin\beta – v)\sin\varphi_2 – \frac{u \cos\beta}{i_{21}} – A \sin\beta $$
The equation of the contact line on the generating plane is therefore:
$$ \begin{cases} \mathbf{r}(u,v) = u\mathbf{e}_1 + v\mathbf{e}_2 \\ \Phi(u,v,\varphi_2) = 0 \end{cases} $$
The velocity of the worm surface \( \mathbf{v}_p^{(1)} \) and the gear surface \( \mathbf{v}_p^{(2)} \) at the contact point, resolved in \( S_p \), are:
| Component | Worm Velocity \( \mathbf{v}_p^{(1)} \) | Gear Velocity \( \mathbf{v}_p^{(2)} \) |
|---|---|---|
| Along \( \mathbf{e}_1 \) | \( v_{px}^{(1)} = Z_2 \cos\varphi_2 \) | \( v_{px}^{(2)} = (r_b – v \sin\beta) i_{21} \) |
| Along \( \mathbf{e}_2 \) | \( v_{py}^{(1)} = (A – r_b \sin\varphi_2 – u \cos\varphi_2) \cos\beta \) | \( v_{py}^{(2)} = u \sin\beta / i_{21} \) |
| Along \( \mathbf{n} \) | \( v_{pz}^{(1)} = (r_b \sin\varphi_2 + u \cos\varphi_2 – A) \sin\beta – v \sin\varphi_2 \) | \( v_{pz}^{(2)} = u \cos\beta / i_{21} \) |
The unit vector along the common normal in the direction of the generating plane is:
$$ \mathbf{a}^{(I)} = \left( \frac{\cos\beta}{i_{21}} – \cos\varphi_2 \sin\beta \right) \mathbf{e}_1 + \sin\varphi_2 \mathbf{e}_2 $$
The first kind of limit function \( \Psi \) and the normal induced curvature \( k_\sigma \) are essential for defining the equivalent contact geometry in the EHL model:
$$ \Psi = \Phi_t + \left( \frac{\cos\beta}{i_{21}} – \cos\varphi_2 \sin\beta \right)(v_{px}^{(1)} – v_{px}^{(2)}) + \sin\varphi_2 (v_{py}^{(1)} – v_{py}^{(2)}) $$
$$ k_\sigma = \frac{ (\mathbf{a}^{(I)})^2 }{ \Psi } $$
The normal velocities of the surfaces are:
$$ v_1 = \mathbf{v}_p^{(1)} \cdot \frac{\mathbf{a}^{(I)}}{\|\mathbf{a}^{(I)}\|}, \quad v_2 = \mathbf{v}_p^{(2)} \cdot \frac{\mathbf{a}^{(I)}}{\|\mathbf{a}^{(I)}\|} $$
The entrainment (rolling) velocity \( U \), a critical parameter in the Reynolds equation, is:
$$ U = \frac{v_1 + v_2}{2} $$
The load per unit length \( w_i \) on the \( i\)-th contacting tooth pair, accounting for load sharing among multiple engaged teeth, is:
$$ w_i = \frac{F_n}{L} = \frac{2 \varepsilon_i T_1}{L d_1 \cos\alpha_n \cos\gamma} $$
where \( F_n \) is the normal force, \( L \) is the contact line length, \( T_1 \) is the input torque on the worm, \( d_1 \) is the worm reference diameter, \( \alpha_n \) is the normal pressure angle, \( \gamma \) is the lead angle, and \( \varepsilon_i \) is the load-sharing coefficient for the \( i\)-th pair. The load-sharing coefficient is determined by the relative stiffness of each contacting pair:
$$ \varepsilon_i = \frac{E_i^s}{E_i^\Sigma}, \quad E_i^s = \frac{1}{(E_1 s_{1i}^3)^{-1} + (E_2 s_{2i}^3)^{-1}}, \quad E_i^\Sigma = \sum_{i=1}^{m} E_i^s $$
where \( E_1, E_2 \) are elastic moduli, \( s_{1i}, s_{2i} \) are instantaneous tooth thicknesses, and \( m \) is the number of simultaneously engaged tooth pairs.
2. Fundamental Equations of Thermal EHL
The contact between the crowned worm and the planar internal gear in this specific worm gear drive is a complex spatial line contact. For EHL analysis, this is simplified to an equivalent contact between an elastic cylinder and a plane. The equivalent radius of curvature \( R \) is the reciprocal of the normal induced curvature: \( R = 1 / k_\sigma \).
The governing equations for the line-contact thermal EHL problem are as follows:
1. Reynolds Equation:
This equation governs the pressure generation within the lubricant film.
$$ \frac{d}{dx}\left( \frac{\rho h^3}{\eta} \frac{dp}{dx} \right) = 12U \frac{d(\rho h)}{dx} $$
with boundary conditions: \( p(x_{in}) = 0 \) and \( p(x_{out}) = dp/dx |_{x_{out}} = 0 \), where \( x_{in} \) and \( x_{out} \) define the computational domain.
2. Energy Equation:
This equation accounts for heat generation due to shear and compression within the film.
$$ \rho c_p U \frac{\partial T}{\partial x} – K \frac{\partial^2 T}{\partial z^2} + \frac{T}{\rho} \frac{\partial \rho}{\partial T} U \frac{\partial p}{\partial x} = \eta \left( \frac{\partial u}{\partial z} \right)^2 $$
The temperature boundary conditions at the worm surface (\( z=0 \)) and gear surface (\( z=h \)) incorporate heat conduction into the solids:
$$ T(x,0) = \frac{K}{\sqrt{\pi \rho_1 c_1 K_1 U_1}} \int_{-\infty}^{x} \frac{\partial T}{\partial z}\bigg|_{(s,0)} \frac{ds}{\sqrt{x-s}} + T_0 $$
$$ T(x,h) = \frac{K}{\sqrt{\pi \rho_2 c_2 K_2 U_2}} \int_{-\infty}^{x} \frac{\partial T}{\partial z}\bigg|_{(s,h)} \frac{ds}{\sqrt{x-s}} + T_0 $$
3. Film Thickness Equation:
This equation describes the shape of the lubricant gap, including elastic deformation of the surfaces.
$$ h(x) = h_0 + \frac{x^2}{2R} – \frac{2}{\pi E’} \int_{x_{in}}^{x_{out}} p(s) \ln|x-s| \, ds $$
where \( h_0 \) is the central film thickness, and \( E’ \) is the effective elastic modulus:
$$ \frac{1}{E’} = \frac{1}{2} \left( \frac{1-\mu_1^2}{E_1} + \frac{1-\mu_2^2}{E_2} \right) $$
4. Viscosity-Pressure-Temperature (VPT) Relation:
The Roelands equation is used to model the dramatic increase in lubricant viscosity with pressure and its decrease with temperature.
$$ \eta(p, T) = \eta_0 \exp\left( (\ln \eta_0 + 9.67) \left[ (1 + 5.1 \times 10^{-9}p)^{Z_0} ( (T-138)/(T_0-138) )^{-S_0} – 1 \right] \right) $$
where \( Z_0 = \alpha / [5.1 \times 10^{-9} (\ln \eta_0 + 9.67)] \), \( S_0 = \beta / [(\ln \eta_0 + 9.67)/(T_0-138)] \), \( \alpha \) is the pressure-viscosity coefficient, and \( \beta \) is the temperature-viscosity coefficient.
5. Density-Pressure-Temperature (DPT) Relation:
$$ \rho(p, T) = \rho_0 \left( 1 + \frac{0.6 \times 10^{-9} p}{1 + 1.7 \times 10^{-9} p} – 0.00065 (T – T_0) \right) $$
6. Load Balance Equation:
The integrated pressure must support the applied load.
$$ w = \int_{x_{in}}^{x_{out}} p(x) \, dx $$
3. Numerical Solution Method
The system of highly nonlinear integral-differential equations is solved numerically. To enhance stability and efficiency, the equations are first non-dimensionalized. Key dimensionless parameters are defined as:
| Parameter | Definition | Parameter | Definition |
|---|---|---|---|
| \( X = x/b \) | Coordinate | \( H = hR/b^2 \) | Film thickness |
| \( P = p/p_h \) | Pressure | \( \bar{W} = w/(E’ R) \) | Load |
| \( \bar{U} = \eta_0 U/(E’ R) \) | Speed | \( G = \alpha E’ \) | Material |
| \( \bar{\eta} = \eta/\eta_0 \) | Viscosity | \( \bar{\rho} = \rho/\rho_0 \) | Density |
| \( \theta = T/T_0 \) | Temperature |
Here, \( b \) is the Hertzian half-width and \( p_h \) is the maximum Hertzian pressure.
The numerical procedure employs the following advanced techniques:
- Finite Difference Method: The dimensionless Reynolds equation is discretized using second-order central differences.
- Multigrid Method (W-cycle): A 6-level grid structure is used to accelerate the convergence of the pressure solution. The finest grid contains 961 nodes. Gauss-Seidel iteration with a relaxation factor \( \omega_p \) is applied on each grid level.
- Multigrid Integration: The elastic deformation integral in the film thickness equation is calculated efficiently using this method.
- Column-by-Column Scanning: The two-dimensional energy equation is solved iteratively using this technique with a relaxation factor \( \omega_T \).
The initial guess for pressure is the Hertzian distribution, for film thickness is the parabolic gap, and for temperature is the ambient temperature \( T_0 \). The solution iterates until the following convergence criteria are met simultaneously:
$$ \frac{\sum |P^{k} – P^{k-1}|}{\sum P^{k}} < 10^{-4}, \quad \frac{|\bar{W}_{calc} – \bar{W}_{input}|}{\bar{W}_{input}} < 10^{-5}, \quad \frac{\sum |\theta^{k} – \theta^{k-1}|}{\sum \theta^{k}} < 10^{-5} $$
4. Influence of Key Design Parameters on TEHL Characteristics
The analysis is conducted for a worm gear drive with the following baseline parameters: center distance \( A = 100 \, \text{mm} \), worm threads \( Z_1 = 1 \), gear teeth \( Z_2 = 63 \), normal module \( m_t = 4.0 \, \text{mm} \), base circle radius \( r_b = 40 \, \text{mm} \), generating plane inclination \( \beta = 0^\circ \). The input power is 2 kW at 1000 rpm. Material and lubricant properties are listed in the table below.
| Component | Elastic Modulus (GPa) | Poisson’s Ratio | Thermal Conductivity (W/m·K) | Specific Heat (J/kg·K) | Density (kg/m³) |
|---|---|---|---|---|---|
| Crown Worm | 206 | 0.30 | 46 | 470 | 7850 |
| Planar Internal Gear | 115 | 0.34 | 109 | 377 | 8760 |
| Lubricant | – | – | 0.14 | 2000 | 890 |
Lubricant: \( \eta_0 = 0.028 \, \text{Pa·s} \), \( \alpha = 22 \, \text{GPa}^{-1} \), \( \beta = 0.476 \, \text{K}^{-1} \), \( T_0 = 313 \, \text{K} \).
The meshing analysis yields multiple instantaneous contact lines for a given worm position. Typically, five lines (I to V) are identified, with Line I corresponding to the tooth pair just entering mesh and Line V to the pair about to leave mesh. The TEHL analysis is performed at characteristic points along these lines. Key performance outputs are the dimensionless central/minimum film thickness \( H_c \), the maximum pressure \( P_{max} \) (including the characteristic EHL pressure spike), and the temperature rises: in the lubricant film mid-layer \( \Delta T_f \), on the worm surface \( \Delta T_1 \), and on the gear surface \( \Delta T_2 \).
General TEHL Behavior:
The results confirm that this planar internal gear enveloping worm gear drive operates in the elastohydrodynamic lubrication regime. A pronounced pressure spike and corresponding film thickness constriction (necking) are observed near the outlet. The TEHL conditions are most severe at the mesh-in point (Line I) and become progressively more favorable towards the mesh-out point (Line V). Specifically, the film thickness increases, and the pressure spike magnitude and lubricant/surface temperature rises decrease significantly from mesh-in to mesh-out. The mid-layer lubricant temperature rise \( \Delta T_f \) is the highest, followed by the worm surface temperature rise \( \Delta T_1 \), and then the gear surface rise \( \Delta T_2 \).
4.1 Effect of Base Circle Radius \( r_b \)
Increasing the base circle radius improves the TEHL performance of this worm gear drive. A larger \( r_b \) effectively increases the equivalent radius of curvature \( R \) at the contact, which is beneficial for film formation. The results show that both the central film thickness \( H_c \) and the pressure spike height increase with \( r_b \). More importantly, the temperature rises \( \Delta T_f, \Delta T_1, \Delta T_2 \) decrease substantially. The effect is more pronounced at the mesh-in position than at mesh-out.
| Parameter Change | Avg. Change in \( H_c \) | Avg. Change in \( \Delta T_f \) | Avg. Change in \( \Delta T_1 \) | Avg. Change in \( \Delta T_2 \) |
|---|---|---|---|---|
| \( r_b \) ↑ by 2 mm | +2.42% | -4.10% | -3.59% | -4.36% |
| \( m_t \) ↑ by 0.25 mm | +2.16% | +36.87% | +19.81% | +22.19% |
| \( \beta \) ↑ by 1° | -0.15% | -0.75% | -0.30% | -0.36% |
4.2 Effect of Normal Module \( m_t \)
The normal module has a complex and significant influence. Increasing \( m_t \) leads to a larger gear size and contact load for a given torque, but also changes the contact geometry and line length. The analysis indicates that while the film thickness \( H_c \) increases slightly with module, the detrimental effects dominate: the pressure spike rises significantly, and, most critically, all temperature rises (\( \Delta T_f, \Delta T_1, \Delta T_2 \)) increase dramatically. This suggests that thermal effects become a major concern in larger-module versions of this worm gear drive. The thermal increase is most severe at the mesh-out point.
4.3 Effect of Generating Plane Inclination Angle \( \beta \)
Introducing a non-zero inclination angle \( \beta \) tilts the plane of the internal gear tooth. This parameter allows for adjustment of the meshing and contact conditions. The TEHL analysis shows that increasing \( \beta \) slightly reduces the film thickness \( H_c \) and the pressure spike. However, it consistently leads to a reduction in all temperature rises across the contact zone. This is a favorable outcome for controlling operating temperatures. The cooling effect, though modest per degree, is most beneficial at the critical mesh-in region where temperatures are highest.
5. Conclusion
The comprehensive thermal elastohydrodynamic lubrication analysis presented in this work provides deep insight into the operational performance of the planar internal gear enveloping crown worm gear drive. The key conclusions are summarized as follows:
- Favorable Lubrication Regime: The transmission inherently operates with effective elastohydrodynamic lubrication. The conditions are more severe at the tooth mesh-in point but improve significantly towards the mesh-out point, indicating a running-in effect within a single engagement cycle.
- Parameter Influence on Film Thickness & Pressure:
- Base Circle Radius (\( r_b \)): Increasing \( r_b \) is beneficial, thickening the EHL film (\( +2.42\% \) avg. per 2 mm).
- Normal Module (\( m_t \)): A larger module also increases film thickness slightly (\( +2.16\% \) avg. per 0.25 mm) but at the cost of a significantly elevated pressure spike.
- Generating Plane Inclination (\( \beta \)): Increasing \( \beta \) causes a very slight reduction in film thickness (\( -0.15\% \) avg. per degree).
- Parameter Influence on Thermal Performance (Critical Finding):
- Base Circle Radius (\( r_b \)): The most effective parameter for reducing operating temperatures. Increasing \( r_b \) lowers lubricant and surface temperature rises by approximately 4% on average per 2 mm increase.
- Normal Module (\( m_t \)): Has a profoundly negative impact on thermal performance. Increasing \( m_t \) leads to drastic increases in temperature rises (over 36% for lubricant mid-layer temperature per 0.25 mm), making thermal management a primary design concern for larger modules.
- Generating Plane Inclination (\( \beta \)): Offers a consistent, though modest, cooling effect, reducing temperature rises by about 0.3-0.75% per degree, which is most valuable at the high-temperature mesh-in region.
Design Implications: The results create a clear trade-off landscape for designers of this type of worm gear drive. To optimize TEHL performance, particularly to minimize the risk of thermal failure and maintain efficiency, the following guidelines are suggested:
– Prioritize selecting a larger base circle radius \( r_b \) within spatial constraints.
– Choose the smallest feasible normal module \( m_t \)* that meets torque and strength requirements to avoid excessive heat generation.
– Consider implementing a positive generating plane inclination angle \( \beta \) as a secondary measure to gain incremental thermal benefits and fine-tune the contact pattern.
This balanced parametric approach, informed by TEHL analysis, is essential for developing reliable, high-performance compact speed reducers based on the planar enveloping principle for advanced applications like robotic joints.