In the field of gear design, internal gears play a critical role in various mechanical systems, particularly in compact transmission units where space efficiency is paramount. As an internal gear manufacturer, we often encounter challenges in accurately predicting the bending strength under multi-tooth meshing conditions, especially for involute internal gear pairs with small tooth number differences. These gears are commonly used in planetary drives, reducers, and other high-precision applications where load distribution and stress analysis are essential for reliability. Traditional methods may not fully account for the complex interactions in multi-tooth engagement, leading to potential overdesign or failure risks. Therefore, we have developed a comprehensive approach to calculate bending strength, incorporating gap analysis, load sharing, and finite element validation. This method not only enhances the design process for internal gears but also provides valuable insights for internal gear manufacturers aiming to optimize performance and durability.
The bending strength of internal gears with small tooth number differences is influenced by multiple factors, including the number of teeth in contact, the distribution of loads among meshing tooth pairs, and the resulting root stresses. In such configurations, the重合度 (degree of overlap) typically ranges between 1 and 2, meaning that both single and double tooth meshing regions exist. This complexity necessitates a detailed analysis of non-theoretical meshing pairs, where gaps between tooth profiles affect load sharing and stress concentrations. Our work builds on existing theories but introduces novel calculations for circumferential and normal gaps, load distribution coefficients, and a systematic approach to determine the number of engaged tooth pairs. By integrating these elements, we aim to provide a robust framework that internal gear manufacturers can adopt to improve the accuracy of bending strength evaluations, ultimately leading to more efficient and reliable gear systems.
To begin, we focus on the calculation of gaps in non-theoretical meshing tooth pairs. In an involute internal gear pair with a small tooth number difference, the external gear and internal gear interact in a way that the circumferential gap increases with distance from the pitch point. Assuming the internal gear is stationary and the external gear rotates clockwise, the meshing process involves an engagement zone where multiple teeth may be in contact simultaneously. For the engagement side, the gap between the external gear tooth profile and the internal gear tooth tip is minimal, while for the disengagement side, the gap between the external gear tooth tip and the internal gear tooth profile is minimal. The circumferential gap, denoted as \( j \), can be derived based on geometric relationships and involute properties.
For the engagement side, the circumferential gap \( j_1 \) is calculated as the arc length between points on the tooth profiles. Let \( \Delta \phi_1 \) represent the gap angle, \( r_o \) the base radius, and \( \alpha \) the pressure angle. The normal gap \( j_{n1} \) is then given by:
$$ j_{n1} = j_1 \cos \alpha $$
Similarly, for the disengagement side, the circumferential gap \( j_2 \) and normal gap \( j_{n2} \) are derived using analogous geometric considerations. The key equations involve the involute function and base circle parameters. For instance, the angle \( \phi_2′ \) for the engagement side can be expressed as:
$$ \phi_2′ = \alpha’ – \frac{\pi}{2z_2} + \text{inv} \alpha_0 $$
where \( \alpha’ \) is the operating pressure angle, \( z_2 \) is the number of teeth on the internal gear, and \( \text{inv} \alpha_0 \) is the involute function of the standard pressure angle. The transformation of these equations allows us to compute the gaps accurately, which is crucial for subsequent load distribution analysis. As internal gear manufacturers, we verify these calculations using solid models and finite element analysis to ensure precision.
Next, we address the positions of tooth pairs along the theoretical line of action. In a multi-tooth meshing scenario, the theoretical meshing point \( K_0 \) can be set at different locations, such as the midpoint of the engagement zone or at the endpoints \( B_2 \) and \( B_1 \). These positions determine whether single or double tooth pairs are engaged. The distance between adjacent meshing points equals the normal base pitch \( p_b = \pi m \cos \alpha \), where \( m \) is the module. By defining the positions relative to \( K_0 \), we can identify non-theoretical meshing points and their corresponding gaps. For example, in the engagement side, the distance \( K’N_2 \) is given by:
$$ K’N_2 = K_0 N_1 + N_1 N_2 – \pi m \cos \alpha $$
This systematic positioning helps in mapping all potential meshing pairs and their interactions, which is essential for accurate load sharing calculations in internal gears.
Load distribution among meshing tooth pairs is a critical aspect of bending strength analysis. When multiple teeth share the load, the distribution coefficients \( \beta_i \) (for \( i = 0, 1, \dots, m-1 \)) determine the fraction of the total normal load \( F_n \) carried by each pair. The model assumes that the sum of deformations and gaps is constant across pairs, leading to the following equation for the load distribution coefficient:
$$ \beta_i = \frac{1}{m} – \frac{\sum_{k=0}^{m-1} j_{nk}}{m F_n / C_m} + \frac{j_{ni}}{F_n / C_m} $$
where \( C_m \) is the meshing stiffness, and \( j_{ni} \) is the normal gap for the \( i \)-th pair. This formula ensures that the load is apportioned based on the relative stiffness and gap sizes. For internal gear manufacturers, accurately determining these coefficients is vital for predicting stress levels and avoiding premature failure. We validate this using finite element simulations, which often reveal that practical factors like rim deformation can alter the ideal distribution, necessitating adjustments in the calculation.
To determine the number of meshing tooth pairs \( m \), we employ an iterative approach. Starting from \( m = 2 \), we calculate the load distribution coefficients and check for non-negative values. The maximum \( m \) that satisfies this condition is selected. This method accounts for the dynamic nature of meshing in internal gears with small tooth number differences, where the number of engaged pairs can vary with load and position. For instance, in our case study with a unit normal load \( F_n / (m b) = 426 \, \text{N} \) (where \( b \) is the face width), we found that \( m \) can be 3 or 4 depending on the position of \( K_0 \). This flexibility is crucial for internal gear manufacturers to adapt designs to specific operating conditions.
The bending strength calculation integrates all the above elements: gap computation, meshing pair identification, load distribution, and root stress evaluation. We use software tools like Mathcad for numerical computations, GearTeq and SolidWorks for modeling, and finite element analysis for validation. A detailed case study illustrates the process. Consider an involute internal gear pair with parameters: module \( m = 2 \, \text{mm} \), face width \( b = 1 \, \text{mm} \), tooth numbers \( z_1 = 80 \) and \( z_2 = 81 \) (one-tooth difference), addendum coefficient \( h_a^* = 0.8 \), and profile shift coefficients \( x_1 = 0 \), \( x_2 = 0.6 \). The重合度 is \( \varepsilon = 1.108 \), and the meshing stiffness is taken as \( C_m = 20 \, \text{N}/\mu\text{m} \cdot \text{mm} \).
First, we compute the circumferential gaps for non-theoretical meshing pairs. The table below summarizes the calculated and measured gaps (in micrometers) for different positions of \( K_0 \):
| Tooth Pair | Engagement Side (j1) | Disengagement Side (j2) | Normal Gap (jn1) | Normal Gap (jn2) |
|---|---|---|---|---|
| Pair 4 | 15.2 | 12.8 | 14.1 | 11.9 |
| Pair 2 | 8.7 | 7.3 | 8.1 | 6.8 |
| Pair 0 | 0.0 | 0.0 | 0.0 | 0.0 |
| Pair 1 | 6.5 | 5.1 | 6.0 | 4.7 |
| Pair 3 | 13.9 | 11.5 | 12.9 | 10.7 |
The measured values, obtained from solid models, align closely with calculations, confirming the accuracy of our gap analysis. This step is fundamental for internal gear manufacturers to ensure proper tooth engagement and minimize wear.
For load distribution, we calculate the coefficients \( \beta_i \) and the corresponding normal loads \( F_{ni} = F_n \beta_i \). Under the given load, the distribution varies with the position of \( K_0 \). The table below shows the calculated and finite element-based measured load distribution (in micrometers of deformation) for different meshing pairs:
| Tooth Pair | K0 at Midpoint (Calc.) | K0 at Midpoint (Meas.) | K0 at B2 (Calc.) | K0 at B2 (Meas.) | K0 at B1 (Calc.) | K0 at B1 (Meas.) |
|---|---|---|---|---|---|---|
| Pair 4 | 0.12 | 0.10 | 0.15 | 0.13 | 0.11 | 0.09 |
| Pair 2 | 0.18 | 0.16 | 0.20 | 0.18 | 0.17 | 0.15 |
| Pair 0 | 0.25 | 0.22 | 0.28 | 0.25 | 0.26 | 0.23 |
| Pair 1 | 0.19 | 0.17 | 0.22 | 0.19 | 0.18 | 0.16 |
| Pair 3 | 0.13 | 0.11 | 0.16 | 0.14 | 0.12 | 0.10 |
The finite element results often show a more uniform distribution due to rim deformation effects, highlighting the importance of considering structural flexibility in real-world applications for internal gear manufacturers.
Finally, we compute the tooth root tensile stress, which is a direct indicator of bending strength. The stress is evaluated for each meshing pair, with the maximum typically occurring on the disengagement side. Using the load distribution data, we apply the root stress formula based on beam theory or finite element analysis. The table below presents the calculated and finite element-based root stresses (in MPa) for the external gear in our case study:
| Tooth Pair | K0 at Midpoint (Calc.) | K0 at Midpoint (Meas.) | K0 at B2 (Calc.) | K0 at B2 (Meas.) | K0 at B1 (Calc.) | K0 at B1 (Meas.) |
|---|---|---|---|---|---|---|
| Pair 4 | 96 | 85 | 154 | 143 | 99 | 85 |
| Pair 2 | 132 | 105 | 173 | 172 | 130 | 105 |
| Pair 0 | 148 | 126 | 220 | 203 | 147 | 126 |
| Pair 1 | 185 | 166 | — | — | 184 | 166 |
| Pair 3 | — | — | — | — | — | — |
The results indicate that multi-tooth meshing leads to varying stress levels, and the maximum stress must be below the allowable limit to ensure safety. For internal gears with small tooth number differences, the convex-concave contact between teeth can result in area contact under load, which may reduce stress concentrations compared to point contact assumptions. Internal gear manufacturers should account for this in design to avoid underestimating strength.
In conclusion, our method provides a systematic approach to calculate the bending strength of internal gears under multi-tooth meshing conditions. By integrating gap calculations, load distribution analysis, and finite element validation, we offer a reliable tool for internal gear manufacturers to optimize designs. The case study demonstrates the method’s accuracy, with close agreement between calculated and measured values. Future work could explore the effects of thermal loads and dynamic conditions, but this framework already represents a significant advancement for the industry. As internal gear manufacturers continue to push the boundaries of efficiency and durability, such detailed analyses will be indispensable for developing next-generation transmission systems.