In recent years, the rapid development of robotics and mechatronics has imposed higher demands on gear transmission systems, particularly those requiring high reduction ratios. Hypoid gears, known for their ability to transmit motion between non-intersecting axes with high efficiency and load capacity, have been adapted to achieve high reduction ratios. These high-reduction ratio hypoid gears, often resembling cone worm gears, retain the advantages of traditional hypoid gears while enabling significant speed reduction and compact design. However, due to the high transmission ratios, wear issues become pronounced, with the pinion often failing before the gear reaches its service life. Understanding and calculating the sliding rate, which is closely related to wear and scuffing, is crucial for optimizing the performance and durability of these gears. This study focuses on developing a method to calculate the sliding rate for high-reduction ratio hypoid gears, providing a theoretical foundation for wear analysis and design improvements.
The design of high-reduction ratio hypoid gears involves complex geometric parameters. Based on the GLEASON hypoid gear tooth blank calculation method, we propose a novel iterative approach to derive the pinion tooth blank parameters from the gear tooth blank parameters. This method ensures accurate determination of key dimensions, such as pitch radii and spiral angles, which are essential for subsequent sliding rate analysis. The basic parameters include the number of teeth for the gear and pinion (e.g., $Z_2$ and $Z_1$), shaft angle $\Sigma$, offset distance $E$, and module $m_n$. For instance, the pitch diameter of the gear $d_2$ is selected based on load capacity requirements, and the face width $b_2$ is limited to ensure structural integrity. The pitch cone angle of the gear $\delta_2$ is approximated using the formula:
$$ \tan \delta_2′ = \frac{\sin \Sigma}{1.2 (z_2 / z_1 + \cos \Sigma)} $$
Subsequently, the pitch radius of the gear $r_2$ is calculated as $r_2 = \frac{1}{2} (d_2 – b_2 \sin \delta_2)$. The initial spiral angle of the pinion $\beta_{10}$ is typically set to around 35 degrees, and an initial enlargement factor $k’$ is introduced to refine the pinion pitch radius $r_1$. Through iterative calculations, parameters such as the pinion pitch cone angle $\delta_1$, offset angle $\epsilon$, and spiral angles $\beta_1$ and $\beta_2$ are determined. The iterative process involves adjusting the enlargement factor until the curvature radius matches the nominal cutter radius $r_0$, ensuring the design meets manufacturing constraints. This approach provides a comprehensive set of tooth blank parameters, as summarized in the following table for two example gear sets with ratios of 3:45 and 5:60.
| Parameter | 3:45 Gear Set | 5:60 Gear Set |
|---|---|---|
| Gear Teeth ($Z_2$) | 45 | 60 |
| Pinion Teeth ($Z_1$) | 3 | 5 |
| Module ($m_n$) | 1.067 | 1.360 |
| Offset Distance $E$ (mm) | 10 | 14 |
| Pressure Angle $\alpha$ (degrees) | 20 | 19 |
| Shaft Angle $\Sigma$ (degrees) | 90 | 90 |
| Gear Spiral Angle $\beta_2$ (degrees) | 39.00 | 55.15 |
| Pinion Spiral Angle $\beta_1$ (degrees) | 67.10 | 31.12 |
| Gear Pitch Cone Angle $\delta_2$ (degrees) | 77.60 | 84.17 |
| Pinion Pitch Cone Angle $\delta_1$ (degrees) | 10.99 | 5.33 |
| Gear Outer Diameter (mm) | 48.01 | 81.65 |
| Pinion Outer Diameter (mm) | 10.41 | 14.10 |
To analyze the sliding rate, we first examine the spatial meshing motion of point-contact gear pairs. Consider three coordinate systems: a fixed system $S$, and two moving systems $S_1$ and $S_2$ attached to the pinion and gear, respectively. The angular velocities of the gears are denoted as $\boldsymbol{\omega}^{(1)}$ and $\boldsymbol{\omega}^{(2)}$, and the translational velocities of the origins are $\boldsymbol{v}_0^{(1)}$ and $\boldsymbol{v}_0^{(2)}$. For a point $M$ on the tooth surface, the position vectors relative to $S_1$ and $S_2$ are $\boldsymbol{r}^{(1)}$ and $\boldsymbol{r}^{(2)}$, respectively. The relative velocity $\boldsymbol{v}^{(12)}$ at point $M$ is derived from the differential motion, leading to the fundamental sliding rate formulas:
$$ \sigma_1 = \frac{ | d_1 \boldsymbol{r}^{(1)} – d_2 \boldsymbol{r}^{(2)} | }{ | d_1 \boldsymbol{r}^{(1)} | } $$
$$ \sigma_2 = \frac{ | d_2 \boldsymbol{r}^{(2)} – d_1 \boldsymbol{r}^{(1)} | }{ | d_2 \boldsymbol{r}^{(2)} | } $$
These equations indicate that the sliding rates $\sigma_1$ and $\sigma_2$ represent the relative sliding between the tooth surfaces of the pinion and gear, which is a key factor in wear and scuffing. For hypoid gears, the sliding rate can be expressed in terms of design parameters by analyzing the motion at the pitch point. We establish a coordinate system at the pitch point $M$ with axes $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, where $\mathbf{i}$ and $\mathbf{j}$ lie in the pitch plane, and $\mathbf{k}$ is perpendicular to it. The angular velocity vectors are given by:
$$ \boldsymbol{\omega}^{(1)} = \omega_1 ( \cos \delta_1 \sin \beta_1 \, \mathbf{i} – \cos \delta_1 \cos \beta_1 \, \mathbf{j} + \sin \delta_1 \, \mathbf{k} ) $$
$$ \boldsymbol{\omega}^{(2)} = \omega_2 ( – \cos \delta_2 \sin \beta_2 \, \mathbf{i} + \cos \delta_2 \cos \beta_2 \, \mathbf{j} + \sin \delta_2 \, \mathbf{k} ) $$
The relative angular velocity $\boldsymbol{\omega}^{(12)} = \boldsymbol{\omega}^{(1)} – \boldsymbol{\omega}^{(2)}$ is then derived. The relative velocity $\boldsymbol{v}^{(12)}$ at the pitch point is found to be parallel to the $\mathbf{j}$-axis, and its magnitude is expressed as:
$$ v^{(12)} = \frac{ \omega_1 L_1 \sin \delta_1 \sin \beta_1 }{ \cos \beta_2 } $$
where $L_1$ and $L_2$ are the distances from the pitch point to the gear axes. The characteristic vector $\boldsymbol{q}$ is calculated as $\boldsymbol{q} = \boldsymbol{\omega}^{(1)} \times ( \boldsymbol{\omega}^{(2)} \times \boldsymbol{r}^{(2)} ) – \boldsymbol{\omega}^{(2)} \times ( \boldsymbol{\omega}^{(1)} \times \boldsymbol{r}^{(1)} )$, which simplifies to:
$$ \boldsymbol{q} = \omega_1 \omega_2 \left[ \sin \delta_1 \sin \delta_2 ( \sin \beta_1 – \sin \beta_2 ) \, \mathbf{i} – \sin \delta_1 \sin \delta_2 ( L_1 \cos \beta_1 + L_2 \cos \beta_2 ) \, \mathbf{j} + \cos \epsilon ( L_1 \sin \delta_1 \cos \delta_2 + L_2 \sin \delta_2 \cos \delta_1 ) \, \mathbf{k} \right] $$
The unit normal vector $\boldsymbol{n}$ to the tooth surface is $\boldsymbol{n} = \pm \cos \alpha \, \mathbf{i} + \sin \alpha \, \mathbf{k}$, where $\alpha$ is the pressure angle, and the sign depends on whether the gear concave side or pinion convex side is considered. The limit pressure angle $\alpha_0$ is given by:
$$ \tan \alpha_0 = \pm \frac{ L_1 \sin \beta_1 – L_2 \sin \beta_2 }{ L_1 \tan \delta_1 + L_2 \tan \delta_2 } $$
Using these relations, the sliding rates for the hypoid gear pair are derived as functions of the design parameters. For the pinion, the sliding rate $\sigma_1$ is:
$$ \sigma_1 = \frac{ v^{(12)} + \left( \boldsymbol{v}^{(12)}, \boldsymbol{\omega}^{(12)}, \boldsymbol{n} \right) + k_v^{(1)} \left( \boldsymbol{n} \cdot \boldsymbol{q} \right) }{ \left( \boldsymbol{v}^{(12)}, \boldsymbol{\omega}^{(12)}, \boldsymbol{n} \right) + k_v^{(1)} \left( \boldsymbol{n} \cdot \boldsymbol{q} \right) } – 1 $$
where $k_v^{(1)}$ is the normal curvature of the pinion tooth surface. A similar expression holds for the gear sliding rate $\sigma_2$. These formulas incorporate the hypoid gear design parameters, enabling direct calculation of sliding rates from tooth blank data.
To validate the derived sliding rate formulas, we applied them to two high-reduction ratio hypoid gear sets: one with a 3:45 ratio and another with a 5:60 ratio. The tooth blank parameters were calculated using the iterative method described earlier, and the sliding rates at the pitch point were computed. For the 3:45 gear set, the absolute sliding rates were $|\sigma_1| = 1.2381$ for the pinion and $|\sigma_2| = 5.1999$ for the gear. For the 5:60 gear set, the values were $|\sigma_1| = 0.6364$ and $|\sigma_2| = 1.7505$. These results indicate that the sliding rate is higher for the pinion than the gear, consistent with practical observations where the pinion experiences more wear. To further verify the accuracy, we compared the results for the 5:60 gear set with an alternative method based on algebraic expressions from literature, which yielded $|\sigma_1’| = 0.6176$ and $|\sigma_2’| = 1.7613$. The close agreement confirms the validity of our approach.
The calculation of sliding rates for hypoid gears with high reduction ratios is essential for predicting wear and optimizing design. The proposed method for determining tooth blank parameters and deriving sliding rate formulas provides a practical tool for engineers. The results demonstrate that the sliding rate at the pitch point is non-zero, indicating relative sliding that contributes to wear. This study lays the groundwork for further research on wear modeling and design optimization of high-reduction ratio hypoid gears, potentially extending to dynamic analysis and lubrication effects. Future work could involve experimental validation and integration with advanced manufacturing techniques to enhance gear performance and longevity.
In summary, this research presents a comprehensive method for calculating the sliding rate of high-reduction ratio hypoid gears. By combining tooth blank design with spatial motion analysis, we have established a link between design parameters and sliding behavior. The derived formulas and computational examples offer valuable insights for improving the durability and efficiency of these gears in demanding applications such as robotics and precision machinery. The iterative parameter calculation method and sliding rate analysis contribute to the broader field of gear theory, enabling more reliable and efficient transmission systems.