Worm gears are widely used in power transmission and motion control applications. However, the trial manufacturing and experimental testing of worm gear teeth are costly, making it highly valuable to accurately assess their feasibility during the design phase. There exist many methods for precise modeling of worm gears. In this study, we adopt CATIA software because it facilitates easy re‑editing during the analysis phase. The geometric models are directly employed for finite element analysis in ANSYS and for rigid‑flexible coupling dynamic analysis in ADAMS. By comparing the computed results, we verify the correctness of the modeling method and the accuracy of the computational outcomes, thereby providing a reference for product development.
Precise Modeling of Worm Gears
Parameters of the Worm Gear Pair
The worm gear pair under investigation is derived from industrial data. The essential geometric and load parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Module m / mm | 1.25 |
| Number of worm starts Z₁ | 1 |
| Number of worm wheel teeth Z₂ | 42 |
| Worm wheel torque T₂ / N·m | 13.923 |
| Center distance a / mm | 36.50 |
| Lead angle γ / (°) | 3.49 |
| Lead P / mm | 5.65 |
| Pressure angle α / (°) | 20.00 |
| Worm pitch diameter d₁ / mm | 20.50 |
| Worm wheel pitch diameter d₂ / mm | 52.50 |
Calculation of Permissible Stresses
Based on the mechanical design handbook, under the rated power of the electric motor, the tooth surface contact stress is computed as:
$$
\sigma_H = Z_E \sqrt{\frac{9400\,T_2}{d_1 d_2^2} K_A K_V K_\beta}
$$
where \(Z_E = 157\ \sqrt{\text{MPa}}\) is the elastic coefficient; \(K_A = 1.5\) is the application factor; \(K_V = 1\) is the dynamic load factor; and \(K_\beta = 1.2\) is the load distribution factor (for varying loads). Substituting the values yields \(\sigma_H = 286\ \text{MPa}\).
The tooth root bending stress under the same condition is:
$$
\sigma_F = \frac{666\,T_2 K_A K_V K_\beta}{d_1 d_2 m} Y_{FS} Y_\beta
$$
where \(Y_{FS} = 3.8\) is the composite form factor, and the helix angle factor is:
$$
Y_\beta = 1 – \frac{\gamma}{120} = 0.97
$$
Thus, \(\sigma_F = 49\ \text{MPa}\). During gear meshing, the tooth flanks experience high contact stresses that may lead to contact fatigue, while the root region is subjected to asymmetric cyclic bending stresses, potentially causing fatigue fracture if the endurance limit is exceeded.
Finite Element Analysis of Worm Gears
Material Properties
The geometric model established in CATIA is imported into ANSYS for finite element preprocessing. The material parameters are given in Table 2.
| Component | Material | Density / kg·m⁻³ | Poisson’s ratio | Elastic modulus / GPa |
|---|---|---|---|---|
| Worm | 42CrMo | 7850 | 0.28 | 212 |
| Worm wheel | QAl9‑4 | 7500 | 0.33 | 116 |
Boundary Conditions
A rotation of 5° around the X‑axis is applied to the worm in the clockwise direction, while all other five degrees of freedom are fixed. A counter‑torque of 13.923 N·m is applied to the worm wheel about the Z‑axis in the counter‑clockwise direction; all other degrees of freedom are constrained.
Finite Element Mesh
Tetrahedral solid elements are used for meshing. The total number of elements for the worm gear pair is 243,741. The meshed model is depicted in the following figure.
Results and Discussion
The contact state of the worm gears shows a strip‑like stress distribution on the worm wheel tooth flank. The maximum contact stress reaches 266 MPa and occurs at the tooth root region. This is a local stress concentration; the contact band typically exhibits stresses between 100 and 250 MPa. The computed maximum stress from ANSYS is 266 MPa, which is in reasonable agreement with the permissible values obtained from the handbook.
ADAMS Rigid Body Dynamic Analysis
In ADAMS, revolute joints are added and the contact between the worm and worm wheel is defined as a collision contact. The continuous dynamic process of impact is described using an equivalent spring‑damper model. The contact force is represented by the step‑function‑based viscous damping model supplied in ADAMS.
According to Hertz contact theory, when the contact area is circular, the indentation \(\delta\) is:
$$
\delta = \frac{a^2}{R} = \sqrt[3]{\frac{9F^2}{16 R E^{*2}}}
$$
The normal contact force during impact is:
$$
F = K\delta
$$
where the gear contact stiffness coefficient is:
$$
K = \frac{4}{3} \sqrt{R} E^{*}
$$
with
$$
\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2},\qquad
\frac{1}{E^{*}} = \frac{1-\mu_1^2}{E_1} + \frac{1-\mu_2^2}{E_2}
$$
Here, \(R_1\) and \(R_2\) are the equivalent radii at the meshing point (approximated by the pitch circle radii), \(\mu_1\) and \(\mu_2\) are the Poisson’s ratios of the worm and worm wheel materials, and \(E_1\) and \(E_2\) are the respective elastic moduli. Substituting the values from Tables 1 and 2 yields:
$$
K = 3.005 \times 10^{11}\ \text{N/mm}^{3/2}
$$
After trial calculations, the damping coefficient is set to \(C = 40\ \text{N·s/mm}\). The dynamic friction coefficient is taken as 0.06 and the static friction coefficient as 0.09, considering lubricated conditions. The force exponent is 1.5, and the maximum damping penetration depth is \(d = 0.1\ \text{mm}\).
To apply the load smoothly, the torque on the worm is defined using the STEP function:
$$
\text{STEP(time, 0, 0, 0.5, 338.8)}
$$
which ramps the torque to 13.923 N·m over 0.5 seconds, avoiding abrupt changes. The simulated meshing force curve shows that the force rapidly increases to 565 N within the first 0.01 s, then slowly rises to a maximum of 587.49 N at 0.19 s due to initial impact, and finally oscillates around 570 N.
ADAMS Rigid‑Flexible Coupling Dynamic Analysis
Modal Analysis
A modal analysis of the worm is performed in HYPERMESH. Modes below 1 Hz are filtered out, and a total of 16 modes are extracted. The first natural frequency is 725 Hz, which is considered effective for the subsequent flexible‑body representation.
Generation of Flexible Body
In HYPERMESH, the worm is defined with material properties, element attributes, RBE2 elements, load collectors, constraints, load steps, and control cards. The model is solved with OptiStruct to produce a modal neutral file (MNF). The MNF file is imported into ADAMS using the “Flex” tool, as illustrated in the settings dialog.
Coupling Setup
Revolute joints are added between the worm, worm wheel, and ground. The contact type is set to “Flexible Body to Solid”. A rotational velocity drive is applied to the worm using the function:
$$
\text{STEP(time, 0, 0, 0.2, 3388d)}
$$
and a counter‑torque of 13.923 N·m is applied to the worm wheel. The solver is configured for a simulation time of 0.35 s with a step size of 20,000.
Results
The rigid‑flexible coupling analysis yields a maximum tooth surface stress of 251 MPa at the root, which is very close to the finite element result of 266 MPa. The deviation is only 5.6%. The meshing force curve from the flexible‑body simulation shows a rapid increase to 573 N within 0.001 s, a peak of 795.49 N at 0.138 s, and then oscillations around 585 N. The larger fluctuations compared to the rigid body analysis are attributed to the more realistic representation of collisions, friction, and inertia effects in the flexible model.
Conclusion
In this study, worm gears are precisely modeled using CATIA and then exported for both finite element analysis in ANSYS and rigid‑flexible coupling dynamic analysis in ADAMS. The tooth contact stress computed from the finite element method is 266 MPa, while the rigid‑flexible coupling simulation gives a maximum stress of 251 MPa, yielding a deviation of merely 5.6%. The meshing force characteristics obtained from the two approaches are also consistent. These results confirm the validity of the modeling methodology and the accuracy of the simulation procedures. The combination of FEA and multibody dynamics provides a reliable tool for predicting the performance of worm gears during the design stage, thereby reducing development costs and shortening the design cycle.