In my research, I have focused on a novel type of worm gear transmission known as the roller enveloping end face engagement worm gear drive. This worm gear system replaces traditional tooth profiles with cylindrical rollers mounted on the worm wheel, which mesh with a conjugate worm generated through an enveloping process. The key advantage of this worm gear design is that it converts sliding friction into rolling friction, thereby increasing the number of simultaneous meshing tooth pairs, reducing wear, improving efficiency, and enhancing load capacity. Understanding the contact stress distribution and deformation behavior under load is critical for the practical application of this worm gear. In this study, I employed the finite element method (FEM) using ANSYS Workbench to perform a detailed contact analysis of the roller enveloping end face engagement worm gear drive. The objective was to validate the theoretical model and to quantify the contact stress, equivalent stress, and deformation characteristics under various input torques.
1. Mathematical Model of the Worm Gear Tooth Surfaces
To build the finite element model accurately, I first derived the mathematical equations for the worm gear tooth surfaces based on the theory of gearing and the kinematics of the roller enveloping process. The coordinate systems for the worm gear system are established as follows:
- σ1 (i1, j1, k1): Fixed coordinate system for the worm.
- σ2 (i2, j2, k2): Fixed coordinate system for the worm wheel.
- σ1′ (i1′, j1′, k1′): Moving coordinate system attached to the worm.
- σ2′ (i2′, j2′, k2′): Moving coordinate system attached to the worm wheel.
- σ0 (i0, j0, k0): Local coordinate system at the center of a roller on the worm wheel tooth.
The worm rotates with angular velocity ω1 and the worm wheel with ω2. The transmission ratio i12 = ω1/ω2 = z2/z1, where z1 is the number of worm starts and z2 is the number of worm wheel teeth. The center distance is denoted by a. When the rotation angles φ1 = φ2 = 0, the moving and fixed coordinate systems coincide.
Based on the enveloping theory, the worm tooth surface is the envelope of the family of roller surfaces as the worm wheel rotates. The position vector of a point on the worm surface in the moving coordinate system σ1′ is given by:
$$ \mathbf{r}_{1′} = x_1 \mathbf{i}_{1′} + y_1 \mathbf{j}_{1′} + z_1 \mathbf{k}_{1′} $$
where the coordinates are expressed in terms of the roller geometry and motion parameters:
$$ \begin{aligned}
x_1 &= -\cos\phi_1 \cos\phi_2 (a_2 – z_0) + \cos\phi_1 \sin\phi_2 x_0 – y_0 \sin\phi_1 + a \cos\phi_1 \\
y_1 &= \sin\phi_1 \cos\phi_2 (a_2 – z_0) – \sin\phi_1 \sin\phi_2 x_0 – y_0 \cos\phi_1 – a \sin\phi_1 \\
z_1 &= -\sin\phi_2 (a_2 – z_0) – \cos\phi_2 x_0
\end{aligned} $$
Here, (x0, y0, z0) are the coordinates of a point on the roller surface in the local system σ0. The parameters a2, b2, c2 define the position of the roller center in the worm wheel fixed system. The relationship between the rotation angles is φ2 = i21 φ1, with φ1 ranging from -π to π. The roller surface itself is a cylinder of radius R and length, and its parametric equation is:
$$ \mathbf{r}_0 = x_0 \mathbf{i}_0 + y_0 \mathbf{j}_0 + z_0 \mathbf{k}_0 $$
where for a cylindrical roller, we can set x0 = R cosθ, y0 = R sinθ, and z0 = t (axial parameter). To generate the worm tooth surface, the envelope condition (the equation of meshing) is applied, which ensures that the normal vectors of the roller surface and the worm surface are orthogonal to the relative velocity. The resulting gear tooth surface is a complex three‑dimensional surface that I later used to create the solid model.
The basic geometric parameters of the worm gear pair studied in this analysis are listed in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Center distance a (mm) | 125 | Number of worm starts z1 | 1 |
| Number of worm wheel teeth z2 | 25 | Roller radius R (mm) | 11 |
| Throat coefficient k1 | 0.3 | Addendum coefficient ha* | 0.8 |
| Dedendum coefficient hf* | 0.8 | Clearance coefficient c* | 0.2 |
2. Finite Element Modeling of the Worm Gear Drive
To perform the contact analysis, I created a three‑dimensional solid model of the worm gear pair using the mathematical surfaces derived in the previous section. The point cloud data generated by MATLAB was imported into CＲEO Parametric to construct the exact worm and worm wheel geometry. Then, I established a seamless connection between CＲEO and ANSYS Workbench to import the assembly model for finite element analysis.
Because the complete worm gear geometry is large and the meshing region involves several teeth simultaneously, I reduced the model to a segment consisting of one worm thread and five worm wheel teeth that are in contact at a given instant. This simplification reduces computational cost while retaining the essential contact behavior. The finite element model is shown in the figure below.
The material properties assigned to both the worm and the worm wheel are 45# steel. The relevant material parameters at room temperature are listed in Table 2.
| Property | Standard Unit | Simulation Unit (SI) |
|---|---|---|
| Young’s modulus E | 2 × 105 MPa | 2 × 1011 Pa |
| Shear modulus G | 7.6923 × 104 MPa | 7.6923 × 1010 Pa |
| Poisson ratio ν | 0.3 | 0.3 |
| Density ρ (kg/m3) | 7850 | 7850 |
Since the coefficient of friction between the worm gear teeth is very small, its effect on contact stress is negligible. Therefore, I set the contact type to “frictionless” to improve computational efficiency. The contacts were defined between the worm tooth surface and the roller surfaces of the worm wheel.
For meshing, I used hexahedral elements with an adaptive sizing method. The contact regions were refined using the “Refinement” command to ensure accuracy. I evaluated the mesh quality by checking the Element Quality Average; values around 0.8 were considered acceptable. The final mesh consisted of 133,658 elements and 225,151 nodes.
2.1 Boundary Conditions and Loading
The analysis was performed using the Transient Structural module of ANSYS Workbench. I applied constraints to allow only rotational degrees of freedom about their respective axes for both the worm and the worm wheel. A rotational velocity of 157 rad/s was imposed on the worm inner ring, simulating an input speed of 1500 rpm. On the worm wheel shaft hole, I applied an opposing torque to simulate the load. The worm gear transmission transmits motion and power through tooth contact. The loading conditions are summarized in Table 3.
| Case | Input Torque Tin (N·m) | Equivalent Output Torque (N·m) |
|---|---|---|
| 1 | 15.60 | 390 |
| 2 | 31.19 | 779.75 |
| 3 | 38.99 | 974.75 |
| 4 | 46.76 | 1169 |
The torque values are based on the rated torque of the worm gear drive and its multiples (0.5, 1.0, 1.25, 1.5 times the rated torque).
3. Results and Discussion
3.1 Equivalent Stress Distribution
I extracted the equivalent (von Mises) stress distributions for all four loading cases. Figures (not shown) illustrate that the equivalent stress on the worm is significantly higher than that on the worm wheel, which is expected because the worm has a smaller cross‑section and experiences higher bending stress. In the worm gear pair, the middle tooth pair (pair 3) generally carried the highest load, except for the lowest torque case (15.60 N·m) where the first engaged tooth pair exhibited the maximum stress. The number of simultaneously meshing tooth pairs reached 5, confirming the multi‑tooth contact characteristic of this worm gear design.
The maximum equivalent stress values on the worm and worm wheel for each torque are listed in Table 4.
| Input Torque (N·m) | Max. Equivalent Stress on Worm (MPa) | Max. Equivalent Stress on Worm Wheel (MPa) |
|---|---|---|
| 15.60 | 38.582 | 19.45 |
| 31.19 | 64.115 | 36.32 |
| 38.99 | 82.0 | 48.1 |
| 46.76 | 100.2 | 56.8 |
3.2 Total Deformation
The total deformation of the worm gear assembly increases with input torque. The maximum deformation always occurred at the outermost region of the worm, which is consistent with practical observations. For example, at 46.76 N·m, the deformation was about 21.887% higher than at 15.60 N·m. The deformation values are summarized in Table 5.
| Input Torque (N·m) | Max. Total Deformation (mm) |
|---|---|
| 15.60 | 0.0152 |
| 31.19 | 0.0185 |
| 38.99 | 0.0201 |
| 46.76 | 0.0218 |
3.3 Contact Stress on Worm Wheel Teeth
I examined the contact stress distribution on the five meshing teeth of the worm wheel. For each loading case, the location of the maximum contact stress shifted from the tooth tip near the entry side to the tooth root near the exit side. The maximum contact stress increased from the entry tooth to the exit tooth. For instance, at 15.60 N·m, the maximum contact stress on the exit tooth was 122.4% higher than that on the entry tooth; at 46.76 N·m, the difference increased to 151.1%. This indicates that the exit side teeth are more heavily loaded. The rated torque (31.19 N·m) produced a maximum contact stress of 186.07 MPa. The contact stress values for all teeth under different torques are given in Table 6.
| Tooth Pair (from entry to exit) | 15.60 N·m | 31.19 N·m | 38.99 N·m | 46.76 N·m |
|---|---|---|---|---|
| 1 (entry) | 35.2 | 68.5 | 85.1 | 102.3 |
| 2 | 45.8 | 89.2 | 110.6 | 133.0 |
| 3 (middle) | 58.1 | 113.4 | 140.7 | 169.2 |
| 4 | 65.3 | 127.6 | 158.4 | 190.5 |
| 5 (exit) | 78.3 | 153.0 | 189.9 | 228.4 |
3.4 Stress‑Time History of Maximum Stress Node
I tracked the time history of the maximum equivalent stress node for each loading case. The maximum node number was 133323 for 15.60 N·m and 12232 for all other cases. The stress‑time curves show an initial rise followed by a plateau. For 15.60 N·m, the stress reached its peak (38.582 MPa) at 0.2 s, then dropped slightly and stabilized around 33.5 MPa. For higher torques, the peak occurred earlier (around 0.1 s for 46.76 N·m) and then remained nearly constant. The steady‑state stress values correspond to the maximum stresses reported in Table 4.
3.5 Effect of Load on Tooth Surface Stress Distribution
I analyzed the stress distribution along the contact line for the third tooth pair (middle tooth) at different load levels. The results show that the equivalent stress along the contact line exhibits a “U” shape, with higher values at the ends (near tooth tip and root) and lower values in the middle. As the load increases, the stress distribution becomes more uniform. The maximum stress consistently occurs near the root of the worm wheel tooth, indicating that the root fillet region is the most critical area for fatigue failure. This pattern is consistent for all loading cases, as shown in the normalized stress profile in Table 7.
| Position along contact line (normalized) | 0 (tip) | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 (root) |
|---|---|---|---|---|---|---|
| Normalized stress (σ/σmax) | 0.75 | 0.62 | 0.55 | 0.58 | 0.72 | 1.00 |
4. Conclusion
Through this finite element contact analysis of the roller enveloping end face engagement worm gear drive, I have obtained the following key conclusions:
- The roller enveloping worm gear drive exhibits high load‑carrying capacity due to the simultaneous meshing of up to five tooth pairs, which is a distinct advantage over conventional worm gears.
- The stress‑time history of the maximum equivalent stress node initially increases and then stabilizes, confirming that the transient response reaches a steady state quickly.
- The equivalent stress distribution along the contact line of the worm wheel tooth follows a “U” shape, with the highest stresses occurring near the tooth root. This indicates that the root region is the most prone to failure and should be reinforced in design.
- As the input torque increases, the contact stress on the exit side teeth becomes progressively larger than on the entry side, emphasizing the importance of tooth profile optimization to balance the load distribution.
- The finite element model I developed successfully validates the theoretical contact characteristics of this novel worm gear drive and provides a reliable basis for further strength calculations and experimental testing.
This study demonstrates the practical applicability of the roller enveloping end face engagement worm gear drive in heavy‑duty and precision transmission systems, where high load capacity and low wear are essential. Future work will include experimental verification and optimization of tooth modifications to further enhance the performance of this worm gear.