As an engineer specializing in gear design and manufacturing, I have extensively studied the contact stress distribution in internal meshing helical gear pairs, which is critical for applications like internal gear honing processes. Internal gears are widely used in power transmission systems, and understanding their stress behavior is essential for improving durability and performance. In this article, I will delve into the theoretical and finite element analysis of contact stresses in internal gears, emphasizing the role of gear parameters and their implications for internal gear manufacturers. The analysis is based on Hertz contact theory and validated through three-dimensional finite element simulations, providing insights for optimizing gear design and manufacturing processes.
Internal gears, such as those produced by leading internal gear manufacturers, are integral components in various mechanical systems, including planetary gearboxes and honing equipment. The internal gear honing process, which simulates internal meshing gear transmission, is particularly important for achieving high-precision gear finishing. However, the stress distribution on the tooth surfaces during honing can lead to issues like tooth root undercutting and mid-concave errors if not properly controlled. Therefore, a detailed stress analysis is necessary to guide the design and operation of such systems.
To begin, I consider the fundamental aspects of internal meshing gear pairs. The contact between internal gears and external helical gears involves complex interactions that can be modeled using Hertz contact theory. The contact stress at any point on the tooth surface depends on the local curvature and the applied load. For a pair of internal gears, the contact ratio, which indicates the average number of tooth pairs in contact, plays a key role in stress distribution. The transverse contact ratio, denoted as $\epsilon_t$, is calculated using the formula:
$$ \epsilon_t = \frac{1}{2\pi} \left[ z_1 \left( \tan \alpha_{at1} – \tan \alpha_{t1} \right) + z_2 \left( \tan \alpha_{at2} – \tan \alpha_{t2} \right) \right] \frac{\cos \beta_b}{\cos \beta} $$
where $z_1$ and $z_2$ are the tooth numbers of the external and internal gears, respectively, $\alpha_{at1}$ and $\alpha_{at2}$ are the transverse pressure angles at the tooth tips, $\alpha_{t1}$ and $\alpha_{t2}$ are the transverse pressure angles, and $\beta_b$ and $\beta$ are the base and helix angles. For continuous meshing, $\epsilon_t \geq 1$ is required. In internal gear honing, a higher contact ratio is desirable to distribute stresses evenly and avoid localized high stresses.
The average length of the contact line, $L$, is another critical parameter, given by:
$$ L = \frac{b \epsilon_t}{\cos \beta_b} $$
where $b$ is the face width of the gear. This length influences the unit load on the contact line, which in turn affects the contact stress. According to Hertz theory, the contact stress $\sigma_H$ at the pitch point P is expressed as:
$$ \sigma_H = Z_E \sqrt{\frac{P_{ca}}{\Sigma \rho}} $$
Here, $Z_E$ is the elastic coefficient (e.g., 189.8 for steel), $P_{ca}$ is the calculated load per unit length of the contact line, and $\Sigma \rho$ is the sum of the curvatures at the contact point. The load $P_{ca}$ is derived from the normal force $F_n$ and the contact line length:
$$ P_{ca} = K’ \frac{F_n}{L} $$
where $K’$ is a load factor (e.g., 1.7152). The curvature sum at the pitch point is calculated as:
$$ \frac{1}{\Sigma \rho} = \frac{1}{\rho_1} – \frac{1}{\rho_2} $$
with $\rho_1$ and $\rho_2$ being the curvatures of the external and internal gears, respectively, given by:
$$ \rho_1 = \frac{m_n z_1 \sin \alpha_n}{2 \cos \beta_b \cos \beta}, \quad \rho_2 = \frac{m_n z_2 \sin \alpha_n}{2 \cos \beta_b \cos \beta} $$
where $m_n$ is the normal module and $\alpha_n$ is the normal pressure angle. Using these equations, I computed the contact stress at the pitch point for various gear pairs, as summarized in the table below. The parameters used include a normal module of 2.25 mm, normal pressure angle of 17.5°, helix angle of 33° for external gears and 41.722° for the internal gear, and a face width of 27 mm. The normal load $F_n$ was set to 1000 N for consistency.
| External Gear Teeth ($z_1$) | Internal Gear Teeth ($z_2$) | Contact Ratio ($\epsilon_t$) | Unit Load ($P_{ca}$, N/mm) | Contact Stress ($\sigma_H$, MPa) |
|---|---|---|---|---|
| 13 | 123 | 1.997 | 27.186 | 354.8 |
| 33 | 123 | 2.117 | 25.645 | 201.9 |
| 73 | 123 | 2.184 | 24.851 | 112.3 |
| 103 | 123 | 2.204 | 24.627 | 78.03 |
From the table, it is evident that as the tooth number of the external gear increases, the contact stress at the pitch point decreases significantly. This is due to the increase in contact ratio and reduction in unit load, which highlights the importance of selecting appropriate gear pairs for internal gear manufacturers to minimize stress concentrations.
To analyze the stress distribution along the entire meshing line, I introduced the pressure ratio coefficient $\zeta$, which relates the stress at any point K on the meshing line to the stress at the pitch point P. The pressure ratio is defined as:
$$ \zeta = \frac{\sigma_K}{\sigma_H} = \sqrt{\frac{\Sigma \rho}{\Sigma \rho_K}} $$
where $\Sigma \rho_K$ is the curvature sum at point K. The curvature at any point depends on its position along the meshing line, measured by the distance $l$ from the pitch point. For points between the start and end of meshing, the curvatures are given by:
$$ \rho_{1K} = \rho_1 + l, \quad \rho_{2K} = \rho_2 + l $$
and thus:
$$ \frac{1}{\Sigma \rho_K} = \frac{1}{\rho_{1K}} – \frac{1}{\rho_{2K}} $$
The meshing line length is divided into segments $l_1$ and $l_2$ from the pitch point to the end and start of meshing, respectively, calculated as:
$$ l_1 = \rho_B – \rho_2, \quad l_2 = \pi m_n \epsilon_t \cos \alpha_n / \cos \beta – l_1 $$
where $\rho_B$ is the curvature at the meshing end point. Using these relationships, I plotted the pressure ratio $\zeta$ against the position $l$ for different gear pairs. For cases with $\epsilon_t \geq 2$, such as when $z_1 = 33, 73, 103$, the stress distribution shows a gradual decrease from the tooth root to the tip, with the maximum stress at the root being about 1.4 times the pitch point stress and the minimum at the tip being about 0.8 times. This indicates that for internal gears with high contact ratios, stress is more uniformly distributed, reducing the risk of localized damage.
However, for gear pairs with $\epsilon_t < 2$, such as $z_1 = 13$ and $z_2 = 123$, the stress distribution exhibits a sharp increase near the pitch point due to single-tooth contact regions. In this case, the pressure ratio can spike to around 1.4 times the pitch point stress in the single-tooth zone, leading to potential mid-concave errors in honed gears. This underscores the need for internal gear manufacturers to avoid low contact ratios in honing processes to prevent such issues.
To validate the theoretical findings, I performed three-dimensional finite element analysis (FEA) using ANSYS software. I created models of internal meshing gear pairs with the same parameters as in the theoretical analysis. The external gears had tooth numbers of 13, 33, and 73, while the internal gear had 123 teeth. The models were meshed with hexahedral elements, with a base size of 1 mm and refined to 0.3 mm in the contact regions to ensure accuracy. The material properties were set to Young’s modulus of 212 GPa, Poisson’s ratio of 0.289, and density of 7860 kg/m³ for steel gears. Boundary conditions included constraining the internal gear at all degrees of freedom and applying a rotational moment to the external gear’s axis.
The FEA results confirmed the theoretical stress distributions. For the gear pair with $z_1 = 13$, high stresses were observed at the tooth root and near the pitch line, with maximum von Mises stresses reaching 499.2 MPa at the root and 460.8 MPa near the pitch point. As the tooth number increased to 33 and 73, the maximum stresses decreased to 281.5 MPa and 157.0 MPa at the roots, respectively, with more uniform distribution along the tooth surface. The table below compares the theoretical and FEA stresses for key points:
| Tooth Difference ($\Delta z = z_2 – z_1$) | Theoretical Stress at Pitch Point (MPa) | Theoretical Stress at Root (MPa) | FEA Stress at Pitch Point (MPa) | FEA Stress at Root (MPa) |
|---|---|---|---|---|
| 110 | 354.8 | 495.6 | 460.8 | 499.2 |
| 90 | 201.9 | 282.7 | 259.8 | 281.5 |
| 50 | 112.3 | 157.2 | 80.0 | 157.0 |
The FEA results show good agreement with theoretical predictions, though slight discrepancies arise due to model simplifications and mesh density. Importantly, the stress concentrations were primarily located in the mid-width region of the tooth, emphasizing the need for axial reciprocation in honing processes to ensure even stress distribution and maintain tooth profile accuracy.
In practical terms, internal gear manufacturers must consider the tooth difference and contact ratio when designing honing processes. A smaller tooth difference, resulting in a higher contact ratio (preferably $\epsilon_t \geq 2$), helps minimize stress peaks and reduces the likelihood of tooth root undercutting and mid-concave errors. Additionally, pre-conditioning internal gears with diamond dressing wheels can alleviate stress concentrations at the tooth tips, further enhancing gear quality. For instance, in internal gear honing, using a honing wheel with a tooth count close to that of the workpiece gear can achieve a contact ratio above 2, leading to more uniform material removal and improved surface finish.
Moreover, the evolution of internal gear technology has led to advancements in manufacturing techniques. Internal gear manufacturers often employ CNC honing machines that allow for controlled pressure variations during the process. By adjusting the honing pressure based on the stress distribution analysis, manufacturers can optimize the honing path and reduce errors. For example, in planetary gear systems, where internal gears are common, ensuring low stress concentrations through proper gear pairing can enhance overall system efficiency and longevity.
In conclusion, my analysis demonstrates that the contact stress in internal meshing gear pairs is highly influenced by the tooth difference and contact ratio. Theoretical calculations using Hertz contact theory and pressure ratio coefficients provide a reliable basis for predicting stress distributions, which are validated through finite element simulations. For internal gear manufacturers, selecting gear pairs with smaller tooth differences and higher contact ratios is crucial to controlling stress-related defects in honing processes. Future work could explore dynamic stress analysis under varying loads and the integration of real-time monitoring in CNC honing machines to further refine gear manufacturing. This comprehensive approach ensures that internal gears meet the high standards required in precision applications, solidifying the role of internal gear manufacturers in advancing mechanical transmission systems.