In our investigation of single roller enveloping face worm gears, we focus on the meshing characteristics that distinguish this transmission from conventional worm gears. The innovative design replaces traditional worm gear teeth with cylindrical rollers that can rotate about their own axes, significantly reducing sliding friction and improving efficiency. Here we present a comprehensive analysis of the contact line distribution, contact line length, roller rotation angle, and sliding speed within this system. Our work builds upon the established meshing theory of roller enveloping worm gears and introduces new concepts such as contact line wrap angle and contact line density to quantify the spatial complexity of the contact lines on the roller cylindrical surface.
1. Coordinate System and Meshing Principle
We established a fixed coordinate system \(\sigma_1(i_1, j_1, k_1)\) and \(\sigma_2(i_2, j_2, k_2)\) for the worm and worm gear, respectively, along with moving coordinate systems \(\sigma_1′(i_1′, j_1′, k_1′)\) and \(\sigma_2′(i_2′, j_2′, k_2′)\) attached to the worm and gear. The roller is mounted on the worm gear, with its rotation axis along \(k_0\) in a local coordinate system \(\sigma_0(i_0, j_0, k_0)\). The angular velocities are \(\omega_1\) and \(\omega_2\), and the transmission ratio is \(i_{12} = Z_2 / Z_1\). The meshing equation for the single roller enveloping face worm gears is derived from the condition that the relative velocity vector is perpendicular to the common normal at the contact point. The relative velocity in the moving frame is expressed as:
\[
\mathbf{V}^{(1’2′)} = V_1^{(1’2′)} \mathbf{e}_1 + V_2^{(1’2′)} \mathbf{e}_2 + V_n^{(1’2′)} \mathbf{n}
\]
The meshing condition yields:
\[
V_n^{(1’2′)} = M_1 \cos\varphi_2 + M_2 \sin\varphi_2 + M_3 = 0
\]
where the coefficients are:
\[
M_1 = \sin\theta (a_2 – u),\quad M_2 = 0,\quad M_3 = -i_{21}\cos\theta (a_2 – u) – A\sin\theta
\]
Here, \(u\) is the roller height parameter, \(\theta\) is the roller angular parameter, \(\varphi_2\) is the worm gear rotation angle, \(A\) is the center distance, and \(a_2\) is the offset of the roller center. Solving the meshing equation gives the functional relationship between \(u\) and \(\theta\):
\[
u = \frac{A\sin\theta + a_2 i_{21}\cos\theta – a_2\sin\theta\cos\varphi_2}{i_{21}\cos\theta – \sin\theta\cos\varphi_2}
\]
This relationship defines the instantaneous contact line on the roller surface for a given \(\varphi_2\).
| Parameter | Symbol | Value |
|---|---|---|
| Center distance | A | 160 mm |
| Number of worm heads | z1 | 1 |
| Number of worm gear teeth | z2 | 25 |
| Throat diameter coefficient | k1 | 0.4 |
| Roller radius | R | 9 mm |
| Worm angular velocity | ω1 | 1 rad/s |
2. Contact Line Analysis
The instantaneous contact line on the roller cylindrical surface is described by the roller surface equation combined with the meshing relation. The length of the i-th contact line is computed via the line integral:
\[
l_i = \int_{\theta_{i1}}^{\theta_{i2}} \sqrt{1 + \left(\frac{dq}{d\theta}\right)^2} d\theta
\]
where \(q(\theta) = f(\theta, \varphi_2)\) from the meshing equation. The contact line wrap angle \(\lambda\) is defined as the central angle subtended by the projection of the contact line onto the roller base plane. It characterizes the spatial complexity of a single contact line and is given by:
\[
\lambda_i = \theta_{i2} – \theta_{i1}
\]
To describe the density of contact lines on the roller surface, we introduce the contact line density \(\varepsilon\), which measures the increment of worm gear rotation \(\varphi_2\) per unit change in roller angle \(\theta\). Its expression is:
\[
\varepsilon = \frac{\Delta\varphi_2}{\Delta\theta} = \left| \frac{d\varphi_2}{d\theta} \right|
\]
Using the parameters from the table above, we computed the contact line length, wrap angle, and density over the entire meshing cycle. The results are summarized in the following table.
| Quantity | Range | Trend |
|---|---|---|
| Contact line length \(l_i\) | 0.0015 mm variation | Nearly constant in first 2/3, sharp increase in last 1/3 |
| Contact line wrap angle \(\lambda\) | 0.1° to 1.3° | Slow increase initially, rapid increase near meshing exit |
| Contact line density at tooth tip | Decreases by ~120 units | Rapid drop at entry, slower later; tip > middle > root |
Our numerical results indicate that the contact lines occupy a very small area on the roller cylindrical surface. The region bounded by the first and last contact lines and the roller top and bottom edges has an area of only 8.148 mm², which is merely 0.88% of the total roller surface area. This implies that each point on the roller surface is repeatedly loaded during meshing, potentially leading to fatigue concerns. The contact line density is highest near the tooth tip, and all positions show a sharp decline as the worm gear rotates, indicating that the contact pattern is more concentrated at the beginning of engagement.
3. Roller Rotation Performance
The roller can rotate about its own axis, which converts sliding into rolling and reduces friction. The rotation performance is evaluated using the self-rotation angle \(\mu_{z0}\), defined as the acute angle between the relative velocity vector and the roller axis. It is calculated as:
\[
\mu_{z0} = \arccos\left( \frac{\mathbf{k}_0 \cdot \mathbf{V}^{(1’2′)}}{|\mathbf{V}^{(1’2′)}|} \right) = \arccos\left( \frac{V_2^{(1’2′)}}{\sqrt{(V_1^{(1’2′)})^2 + (V_2^{(1’2′)})^2}} \right)
\]
Our analysis shows that \(\mu_{z0}\) exceeds 89° throughout the meshing cycle, indicating that the roller rotates very efficiently. The self-rotation angular velocity \(\omega_0\) of the roller is given by:
\[
\omega_0 = \frac{E}{R}, \quad E = \sin\theta\, E_1 – \cos\theta\, E_2
\]
\[
E_1 = a_2 i_{21} – i_{21} u – R \sin\varphi_2 \sin\theta, \quad
E_2 = R \sin\varphi_2 \cos\theta + \cos\varphi_2 (u – a_2) + A
\]
We computed \(\omega_0\) for different roller heights and worm gear rotation angles. The self-rotation angular velocity starts high (27.5–30 rad/s) at meshing entry and gradually decreases to about 7.5 rad/s at exit. The smaller the roller height (closer to tooth tip), the faster the decrease. To quantify the overall rotation, we defined the average self-rotation angular velocity over the roller height:
\[
\omega_{0u} = \frac{1}{u_2 – u_1} \int_{u_1}^{u_2} \omega_0(u) \, du
\]
Using numerical integration in Matlab, we obtained \(\omega_{0u}\) as a function of \(\varphi_2\). The total rotation angle of the roller during one meshing period is:
\[
\theta_z = \int_{t_1}^{t_2} \omega_{0u}(t) \, dt
\]
Substituting the parameters, we found that the roller rotates by 749.8523 rad, which is equivalent to 119.3427 full revolutions. This means that during a single engagement of a worm gear tooth, the roller spins over 119 times, repeatedly distributing the contact load over its entire cylindrical surface.
| Parameter | Value |
|---|---|
| Self-rotation angle \(\mu_{z0}\) | > 89° (close to ideal 90°) |
| Self-rotation angular velocity range | 7.5 – 30 rad/s |
| Total rotation angle per meshing cycle | 749.8523 rad (≈119.3 rev) |
4. Sliding Speed Comparison
One of the key advantages of roller enveloping worm gears is the reduction of sliding. In our design, the sliding speed at the meshing point is only the component of the relative velocity along the \(\mathbf{e}_2\) direction, because the component along \(\mathbf{e}_1\) drives the roller rotation and does not contribute to sliding. The sliding speed is:
\[
V_h = V_2^{(1’2′)} = R \cos\varphi_2 \sin\theta – R i_{21} \cos\theta
\]
For comparison, we also analyzed the case where the roller is fixed (non-rotating), in which the total relative speed becomes the sliding speed:
\[
V_h^{\text{fixed}} = \sqrt{(V_1^{(1’2′)})^2 + (V_2^{(1’2′)})^2 + (V_n^{(1’2′)})^2}
\]
Our numerical results reveal a dramatic difference, as shown in the table below.
| Condition | Sliding speed range (mm/s) | Maximum value |
|---|---|---|
| Roller free to rotate (proposed design) | 0.2 – 0.95 | 0.95 |
| Roller fixed (traditional equivalent) | 55 – 265 | 265 |
With the roller rotating, the sliding speed is only about 0.36% of that in the fixed condition. Even the maximum sliding speed in the rotating case (0.95 mm/s) is merely 1.7% of the minimum sliding speed in the fixed case (55 mm/s). This outstanding reduction in sliding directly translates to lower friction, less heat generation, and higher efficiency in the worm gears.
5. Conclusion
Our comprehensive analysis of single roller enveloping face worm gears has revealed several important characteristics. The contact lines on the roller cylindrical surface are confined to a very small region (0.88% of the total surface), yet the roller makes over 119 revolutions during one meshing cycle, ensuring that load distribution is frequent but concentrated. The self-rotation angle remains above 89°, indicating excellent rotation performance. The sliding speed is reduced to negligible levels compared to a fixed-roller design, confirming the superior tribological behavior of this type of worm gears. These findings provide a solid foundation for further studies on contact strength, fatigue life, and lubrication optimization of roller enveloping worm gear transmissions.