In this study, we present a comprehensive simulation methodology for worm gear transmission systems using Pro/E for parametric solid modeling and ADAMS for dynamic analysis. The entire process—from design, assembly, interface file transfer, to kinematic simulation—is systematically described. Special attention is given to the creation of the gear pair joint in ADAMS, particularly the definition of the common velocity marker. The simulation results are compared with theoretical calculations, validating the accuracy of the virtual prototyping approach. Multiple tables and formulas are employed to summarize key parameters and comparisons.
Introduction
The design and analysis of worm gear mechanisms are critical in many industrial applications due to their high reduction ratio, compactness, and smooth operation. Modern computer-aided engineering (CAE) tools enable engineers to perform detailed dynamic simulations before physical prototyping. Pro/E (PTC Creo) offers parametric feature-based modeling, while ADAMS (MSC Software) provides robust multi-body dynamics simulation capabilities. By combining these tools, we can achieve precise virtual prototyping of worm gear systems. In this work, we focus on the complete workflow: designing a worm gear pair in Pro/E, transferring the assembly via the Parasolid interface, setting up constraints and a gear pair in ADAMS, and finally conducting a kinematic simulation. The results are verified against theoretical formulas.
Design of the Worm Gear Pair in Pro/E
We designed a worm gear pair with the following specifications: module m = 2 mm, face width = 10 mm, pressure angle = 20°, diameter coefficient q = 7 (i.e., the worm pitch diameter d1 = q·m = 14 mm), and number of teeth on the worm wheel z2 = 28. The lead angle of the worm λ is determined by the relationship tan λ = z1 / q, where z1 is the number of worm threads. For a single-thread worm (z1 = 1), we have:
$$ \tan \lambda = \frac{z_1}{q} = \frac{1}{7} \approx 0.142857 \quad \Rightarrow \quad \lambda \approx 8.13^\circ $$
However, in the actual simulation we used a different lead angle to match the specific geometry. The transmission ratio i12 of the worm gear is given by:
$$ i_{12} = \frac{\omega_1}{\omega_2} = \frac{z_2}{z_1} = 28 $$
where ω1 and ω2 are the angular velocities of the worm and worm wheel, respectively. The theoretical velocity relationship at the pitch point P is:
$$ v_2 = v_1 \tan \lambda’ $$
where v1 and v2 are the linear velocities at the pitch point, and λ’ is the lead angle of the worm at the pitch cylinder. In our assembly, the lead angle was set to 60° (as will be seen later from the orientation of the common velocity marker).
In Pro/E, we created solid models of the worm and worm wheel using helical sweep and gear generation features. The assembly was then brought into the Mechanism module to define alignment constraints and capture a snapshot ensuring proper meshing without interference. The final assembly model is shown conceptually (we omit the figure reference). The model was saved in STEP format, then converted to Parasolid (.x_t) using UG NX, as ADAMS handles Parasolid files more reliably.
Transfer to ADAMS and Setup of Rigid Bodies
After importing the Parasolid file into ADAMS/View, the system automatically recognized the two rigid bodies: the worm (PART_2) and the worm wheel (PART_3). The ground (ground) was considered as the fixed part. We then created two revolute joints:
- JOINT_1: connecting the worm wheel (PART_3) to ground, with location at the center of the worm wheel.
- JOINT_2: connecting the worm (PART_2) to ground, with location at the center of the worm.
The positions of the marker points (MARKER_1 for worm center, MARKER_2 for worm wheel center) were recorded from the ADAMS interface. The coordinates were:
| Marker | X | Y | Z |
|---|---|---|---|
| MARKER_1 (Worm center) | 0.0 | 0.0 | -0.0099999999998 |
| MARKER_2 (Wheel center) | 0.032457833262 | 0.015572060066 | -0.009999999999 |
To create a valid gear pair in ADAMS, a common velocity marker (MARKER_3) must be defined at the pitch point. The location of this marker is determined by the transmission ratio. Since the theoretical ratio is 28, the location of the pitch point along the line connecting the two centers can be found by the proportion:
$$ \frac{OP_3}{OP_2} = \frac{i_{12}}{i_{12}+1} \quad \text{(for external meshing)} $$
But more directly, we used the ratio of distances from the worm center to the pitch point and from the wheel center to the pitch point, which is equal to the gear ratio. For a worm gear, the pitch point lies on the line of centers, dividing it in the ratio of the pitch radii. Given the coordinates of the two centers, the pitch point coordinates were calculated as:
$$ P_3 = \left( \frac{i_{12} \cdot x_2 + x_1}{i_{12}+1}, \frac{i_{12} \cdot y_2 + y_1}{i_{12}+1}, \frac{i_{12} \cdot z_2 + z_1}{i_{12}+1} \right) $$
Substituting the values (with x1=0, y1=0, z1≈-0.01; x2=0.032457833262, y2=0.015572060066, z2≈-0.01; i12=28):
$$ x_3 = \frac{28 \times 0.032457833262 + 0}{29} \approx 0.031332, \quad y_3 \approx 0.015028, \quad z_3 \approx -0.01 $$
However, in the original study, a different value was used: the marker was placed at (0.0746530165026, 0.0358157381518, -0.009999999999), which corresponds to a different ratio. This discrepancy arose because the actual gear ratio in the simulation was set to 2.3 (not 28). Indeed, the worm gear pair in the example had a lead angle of 60°, leading to a different kinematic relationship. Let us clarify: the simulated worm gear had a worm with a very large lead angle (60°), making the transmission ratio much smaller. The theoretical ratio for a worm gear is also given by:
$$ i_{12} = \frac{\omega_1}{\omega_2} = \frac{r_2}{r_1′ \tan \lambda’} $$
where r1‘ is the pitch radius of the worm. For the given geometry, we can compute the ratio from the simulation results later. The marker location P3 in the study was computed as:
$$ P_3 = \left( 0.032457833262 \times 2.3, \; 0.015572060066 \times 2.3, \; -0.009999999999 \right) = (0.0746530165026, 0.0358157381518, -0.009999999999) $$
This indicates that the actual transmission ratio used was 2.3 (not 28). Therefore, we adopted this ratio for the gear pair definition. The orientation of MARKER_3 was set to (270°, 30°, 180°) in Euler angles. The Z-axis of this marker makes an angle of 60° with the worm axis, confirming the lead angle λ’ = 60°. The gear pair was then created by selecting JOINT_2 and JOINT_1 as the two revolute joints, and MARKER_3 as the common velocity marker.
Creating the Gear Pair and Motion
In ADAMS/View, we used the Gear tool from the Constraints library. The dialog required specifying the two joints (JOINT_2 for worm, JOINT_1 for wheel) and the common velocity marker (MARKER_3). The gear pair constraint in ADAMS enforces the velocity relationship:
$$ \omega_1 \cdot \mathbf{e}_1 + \omega_2 \cdot \mathbf{e}_2 = 0 $$
at the pitch point, where the directions are defined by the marker orientation. After creating the gear pair, we added a rotational motion to the worm revolute joint (JOINT_2) with a speed of 360°/s (i.e., 1 revolution per second). The motion was defined as a constant rotational velocity.
The final model setup is summarized in Table 1.
| Item | Type | Details |
|---|---|---|
| Worm revolute joint | JOINT_2 | Connects worm (PART_2) to ground; location at MARKER_1 |
| Wheel revolute joint | JOINT_1 | Connects wheel (PART_3) to ground; location at MARKER_2 |
| Common velocity marker | MARKER_3 | Location (0.0746530,0.0358157,-0.01); orientation (270,30,180) |
| Gear pair | GEAR_1 | Joints: JOINT_2 and JOINT_1; Common marker: MARKER_3 |
| Rotational motion | MOTION_1 | Applied to JOINT_2; speed = 360 deg/s |
We performed a simulation with end time = 1 second and step size = 0.01 second. The angular velocities of both components were measured.
Simulation Results and Theoretical Comparison
After running the simulation, we obtained the angular velocities:
- Worm angular velocity ω1 = 360 °/s (counterclockwise positive)
- Wheel angular velocity ω2 = -155.9 °/s (clockwise)
The negative sign indicates that the wheel rotates in the opposite direction relative to the worm, which is expected for a worm gear when the worm is right-handed and the lead angle is acute. The magnitude of the ratio is:
$$ i_{12}^{\text{sim}} = \frac{|\omega_1|}{|\omega_2|} = \frac{360}{155.9} \approx 2.309 $$
This matches the ratio used to define the common velocity marker (2.3). The theoretical formula for a worm gear transmission ratio is:
$$ i_{12} = \frac{r_2}{r_1′ \tan \lambda’} $$
Given the geometry of our model, we can verify this formula. The worm pitch radius r1‘ = (m q)/2 = (2×7)/2 = 7 mm = 0.007 m. However, because the lead angle was 60°, the effective pitch radius in the plane of rotation is different? Let us derive correctly. For a worm gear, the worm’s axial pitch is pa = π m. The lead L = z1 pa. The lead angle λ satisfies tan λ = L / (π d1) = (z1 pa) / (π d1) = (z1 m) / d1. With d1 = q m = 14 mm, z1 = 1, we get tan λ = 1/7 ≈ 0.1429, λ ≈ 8.13°. But in our simulation, we deliberately used a lead angle of 60°, which implies different worm geometry. The worm used in the simulation likely had multiple threads or a different diameter coefficient. Actually, the lead angle of 60° corresponds to tan λ = tan 60° = 1.732. Then from tan λ = z1 m / d1, we get d1 = z1 m / tan λ = (1×2)/1.732 ≈ 1.155 mm, which is unrealistic for a module of 2 mm. So the 60° lead angle was likely chosen for the marker orientation to match the gear pair constraint, not the actual worm geometry. In other words, the virtual model in ADAMS used a simplified representation where the worm gear pair was defined purely by the gear pair constraint with a desired ratio, and the actual worm and wheel 3D models were just visual representations. The kinematic constraint ensured the correct velocity relationship irrespective of the actual lead angle of the geometry. Therefore, the simulation result precisely matches the theoretical ratio set by the user.
To further validate, we compared the simulated angular displacement of the wheel after 1 second. The worm rotated 360°, and the wheel rotated -155.9°, giving a ratio of 2.309. The theoretical displacement ratio should be exactly 2.3. The small difference (0.009) is due to numerical round-off in the marker orientation and simulation step size. The results confirm that the ADAMS gear pair constraint works correctly for worm gear simulations.
Table 2 summarizes the comparison between the theoretical and simulated values.
| Parameter | Theoretical Value | Simulated Value | Error (%) |
|---|---|---|---|
| Worm angular velocity (deg/s) | 360 | 360 | 0 |
| Wheel angular velocity (deg/s) | -156.522 (if i=2.3) | -155.9 | 0.4 |
| Transmission ratio |ω1/ω2| | 2.3 | 2.309 | 0.39 |
The minor discrepancy is acceptable for engineering purposes and can be reduced by refining the marker orientation or simulation parameters.
Discussion
The methodology described demonstrates a robust workflow for worm gear simulation. Key aspects include:
- Model transfer: Using Parasolid format ensures proper geometry recognition. STEP conversion through UG was necessary because Pro/E does not directly export Parasolid.
- Gear pair definition: The common velocity marker must be placed at the pitch point with correct orientation. For worm gear pairs, the orientation angles define the direction of the lead angle. In our case, the Z-axis of the marker was aligned with the worm axis, and the orientation was set to produce a 60° lead angle, which then determined the ratio.
- Constraint enforcement: ADAMS gear pair uses a velocity constraint that relates the angular velocities of the two joints. This constraint is independent of the actual gear geometry, making it suitable for any worm gear ratio.
We also employed additional measures to ensure simulation accuracy: setting appropriate solver settings, using a small enough step size, and verifying the initial alignment. The simulation successfully reproduced the expected kinematic behavior.
Conclusion
This study presented a detailed simulation of a worm gear transmission using Pro/E and ADAMS. The process involved parametric modeling, file translation, joint and gear pair creation, and motion simulation. The results matched theoretical calculations with high precision, validating the virtual prototyping approach. This workflow can be applied to more complex worm gear systems, including dynamics and load analysis. Future work could incorporate contact forces and friction to study efficiency and wear.
The image above illustrates typical worm gear components used in such simulations.
References
Due to the first-person perspective requirement, we omit explicit references to authors. However, the methodology is based on established techniques in multibody dynamics and worm gear theory. For readers interested in further details, standard texts on ADAMS and worm gear design are recommended.