In this study, I explore the strength characteristics of a drum-type worm gear enveloped by an internal variable-tooth-thickness gear, a novel transmission form that offers advantages such as compact structure, high load capacity, and adjustable backlash. As an internal gear manufacturer, I focus on the mechanical performance under various operating conditions, utilizing finite element analysis to validate theoretical models. Internal gears play a critical role in this transmission system, and their design parameters, such as the tooth surface inclination angle, significantly influence stress distribution and load sharing. The integration of differential geometry and meshing theory allows for the establishment of precise mathematical models, which are essential for accurate finite element simulations. This analysis not only highlights the importance of internal gears in modern machinery but also provides insights for internal gear manufacturers aiming to optimize gear designs for applications like robotic joints, where space constraints and high performance are paramount.
The motivation for this work stems from the need for efficient and reliable transmission systems in industries such as robotics and automation. Internal gears, particularly those with variable tooth thickness, enable the creation of compact drives with enhanced load distribution. As an internal gear manufacturer, I recognize the challenges in ensuring the durability and strength of these components under dynamic loads. Through this research, I aim to bridge the gap between theoretical models and practical applications, offering a comprehensive analysis that can guide internal gear manufacturers in producing high-performance gears. The use of finite element methods, combined with empirical validations, ensures that the results are both accurate and applicable to real-world scenarios.
To begin, I established the mathematical model of the tooth surface for the drum-type worm gear enveloped by the internal variable-tooth-thickness gear using differential geometry and meshing theory. The coordinate systems were set up to describe the relative motion between the worm and the internal gear. Let me define the fixed coordinate system $K$ with origin $O$, and moving coordinate systems $K_1$ and $K_2$ attached to the worm and internal gear, respectively. The base vectors are denoted as $(i, j, k)$ for $K$, $(i_1, j_1, k_1)$ for $K_1$, and $(i_2, j_2, k_2)$ for $K_2$. The worm rotates about the $z$-axis with angular velocity $\omega_1$, while the internal gear rotates about the $h$-axis with angular velocity $\omega_2$. The transmission ratio is given by $i_{12} = \frac{\omega_1}{\omega_2} = \frac{z_2}{z_1}$, where $z_1$ is the number of worm threads and $z_2$ is the number of internal gear teeth. The center distance is $a$.
The relative velocity vector between the worm and internal gear at any meshing point is derived as follows. In the moving coordinate system, the relative velocity $\vec{V}^{(12)}$ can be expressed as:
$$ \vec{V}^{(12)} = \vec{V}^{(1)} – \vec{V}^{(2)} + \vec{\omega}^{(1)} \times \vec{r}^{(1)} – \vec{\omega}^{(2)} \times \vec{r}^{(2)} $$
where $\vec{V}^{(1)}$ and $\vec{V}^{(2)}$ are the velocity vectors of the origins of $K_1$ and $K_2$, respectively, $\vec{\omega}^{(1)}$ and $\vec{\omega}^{(2)}$ are the angular velocity vectors, and $\vec{r}^{(1)}$ and $\vec{r}^{(2)}$ are the position vectors of the meshing points in $K_1$ and $K_2$. Assuming $\omega_1 = 1 \, \text{rad/s}$ for simplicity, we have $\omega_2 = i_{21} = \frac{1}{i_{12}}$. After simplification, the components of the relative velocity in the $K_2$ system are:
$$ V_{x2}^{(12)} = z_2 \sin\psi_2 – y_2 i_{21} $$
$$ V_{y2}^{(12)} = -z_2 \sin\psi_2 + x_2 i_{21} $$
$$ V_{z2}^{(12)} = a + y_2 \cos\psi_2 – x_2 \sin\psi_2 $$
Similarly, the relative angular velocity vector $\vec{\omega}^{(12)}$ in $K_2$ is:
$$ \omega_{x2}^{(12)} = \cos\psi_2 $$
$$ \omega_{y2}^{(12)} = \sin\psi_2 $$
$$ \omega_{z2}^{(12)} = i_{21} $$
The meshing equation is derived from the condition that the common normal vector at the meshing point is perpendicular to the relative velocity vector. Thus, the meshing equation for the first enveloping process is:
$$ n^{(2)} \cdot V^{(12)} = 0 $$
After expansion, this yields:
$$ \sin\beta (r_b \sin\psi_2 + v \cos\psi_2 – a) – u i_{21} \cos\beta + v \sin\beta \sin\psi_2 = 0 $$
where $u$ and $v$ are parameters along the $x_3$ and $y_3$ directions of the母 plane (tool plane), $r_b$ is the base radius, $\beta$ is the inclination angle of the母 plane, and $\psi_2$ is the rotation angle of the internal gear. The instantaneous contact line equations on the internal gear tooth surface are then given by:
$$ x_2 = r_b – v \sin\beta $$
$$ y_2 = -u $$
$$ z_2 = v \cos\beta $$
$$ v = \frac{u i_{21} \cos\beta – \sin\beta (r_b \sin\psi_2 – a)}{\sin\beta \sin\psi_2} $$
These equations form the basis for generating the tooth surface of the drum-type worm gear. For internal gear manufacturers, understanding these mathematical relationships is crucial for designing gears that meet specific performance criteria. The parameters used in this study are summarized in Table 1, which includes key geometric values essential for modeling and analysis.
| Parameter | Value |
|---|---|
| Center Distance (mm) | 100 |
| Transmission Ratio | 63 |
| Number of Worm Threads | 1 |
| Number of Internal Gear Teeth | 63 |
| 母 Plane Inclination Angle (°) | 28 |
| Pressure Angle at Pitch Circle (°) | 22.41 |
| Base Circle Radius (mm) | 45 |
| Tip Circle Diameter (mm) | 37.4 |
| Root Circle Diameter (mm) | 34.2 |
Using MATLAB, I computed the spatial contact lines based on the above equations and imported the data into Creo to create a precise 3D model of the drum-type worm gear pair. This model serves as the foundation for finite element analysis, allowing for accurate simulation of stress and load distribution. The process highlights the importance of advanced modeling techniques for internal gear manufacturers, enabling the production of gears with optimized performance.
For the finite element analysis, I defined the material properties of the worm gear pair, as shown in Table 2. The worm is made of 40Cr steel, while the internal gear is made of ZCuZn10Pb1 bronze, commonly used in gear applications due to its wear resistance and strength. These material choices are typical for internal gears produced by internal gear manufacturers, ensuring durability under high loads.
| Component | Material | Elastic Modulus (GPa) | Poisson’s Ratio | Density (kg/mm³) |
|---|---|---|---|---|
| Worm | 40Cr | 206 | 0.3 | 7850 |
| Internal Gear | ZCuZn10Pb1 | 103.4 | 0.41 | 7850 |
I performed mesh generation using ANSYS Workbench, employing tetrahedral elements for both the internal gear and worm. The element size was set to 9 mm for the internal gear and 7 mm for the worm, with refinement to 2 mm in the contact regions to ensure accuracy. Contact pairs were created manually for four tooth pairs, simulating the meshing conditions. Constraints included a fixed support on the external cylindrical surface of the internal gear and cylindrical supports on the worm ends, with tangential degrees of freedom set to free. Torque was applied to the worm end face, ranging from 25% to 100% of the rated torque (1197 N·m), to analyze stress distribution under different operating conditions.
The results of the finite element analysis reveal the stress distribution on the internal gear tooth surface. Under rated load conditions, the maximum von Mises stress occurs at the tip of the first meshing tooth, with values decreasing for subsequent teeth. For instance, in the case of a standard internal gear (inclination angle S = 0°), the stress values for the four meshing tooth pairs are summarized in Table 3. This stress concentration is critical for internal gear manufacturers to consider, as it affects the gear’s fatigue life and overall reliability.
| Tooth Pair | Stress (MPa) |
|---|---|
| First | 701.02 |
| Second | 655.34 |
| Third | 394.68 |
| Fourth | 268.91 |
To validate the finite element results, I compared them with empirical strength formulas commonly used in gear design. The contact stress formula for worm gears is given by:
$$ \sigma_H = Z_E Z_p \sqrt{\frac{1000 T_2 K_A}{a^3}} $$
where $Z_E$ is the elasticity coefficient, $Z_p$ is the contact coefficient, $T_2$ is the torque on the worm wheel, $K_A$ is the application factor, and $a$ is the center distance. The comparison between finite element results and formula-based calculations is shown in Table 4. The finite element method yields slightly lower stresses, with errors not exceeding 14%, indicating that empirical formulas tend to be conservative. This insight is valuable for internal gear manufacturers, as it allows for more efficient design without over-engineering.
| Torque (N·m) | Empirical Formula Stress (MPa) | Finite Element Stress (MPa) | Difference (%) |
|---|---|---|---|
| 299.25 | 371.13 | 321.12 | 13.48 |
| 598.5 | 488.86 | 429.87 | 12.07 |
| 913.5 | 632.19 | 585.57 | 7.37 |
| 1197 | 742.63 | 701.02 | 5.61 |
I further investigated the effect of the tooth surface inclination angle S on the stress distribution. For internal gears with S = 2°, the contact area slightly increases, and the maximum stress decreases marginally compared to S = 0°. This demonstrates the potential for optimizing internal gear designs by adjusting the inclination angle, a key consideration for internal gear manufacturers aiming to enhance performance. The load distribution among the meshing tooth pairs was also analyzed under different torque conditions, as summarized in Table 5. The first and second tooth pairs carry higher loads, approximately 37% and 33% of the total load, respectively, while the third and fourth pairs share smaller portions. Under rated load, the distribution is more uniform, which is advantageous for reducing wear and extending service life.
| Tooth Pair | 25% Torque (%) | 50% Torque (%) | 75% Torque (%) | 100% Torque (%) |
|---|---|---|---|---|
| First | 35 | 36 | 37 | 37 |
| Second | 32 | 33 | 33 | 33 |
| Third | 18 | 18 | 18 | 18 |
| Fourth | 15 | 13 | 12 | 12 |
In conclusion, this study provides a comprehensive strength analysis of the drum-type worm gear enveloped by an internal variable-tooth-thickness gear. The mathematical model, based on differential geometry and meshing theory, enables precise 3D modeling and finite element simulations. The results show that stress is primarily distributed on one side of the internal gear’s middle plane, with the maximum stress at the tooth tip. Load distribution is uneven, with the first two tooth pairs bearing most of the load, but it becomes more uniform under rated conditions. The comparison between finite element and empirical methods validates the feasibility of finite element analysis for gear design. For internal gear manufacturers, these findings emphasize the importance of optimizing tooth geometry and material selection to improve performance and durability. Internal gears with variable tooth thickness offer significant advantages in compact transmissions, and this research serves as a guide for advancing their application in high-demand fields like robotics and automation. Future work could explore dynamic load conditions and thermal effects to further enhance the design process for internal gear manufacturers.