In modern mechanical systems, the demand for high-performance gear transmissions has led to the development of advanced spur gears with improved characteristics. Traditional spur gears, such as straight-cut and helical gears, often fall short in meeting the requirements of high-speed and heavy-duty applications due to issues like vibration, noise, and limited load capacity. This study focuses on a specific type of spur gear known as the variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear, which features a unique tooth geometry that enhances meshing performance. The tooth profile in the central section is an ideal involute curve, while parallel sections exhibit hyperbolic envelopes, and the tooth trace follows a circular arc. This design offers advantages such as high contact ratio, self-aligning capability, absence of axial forces, and insensitivity to installation errors, making it suitable for demanding environments.
The primary objective of this research is to improve the load-bearing capacity and reduce vibration and noise in spur gear systems through tooth surface modification. A method involving inclined cutter milling is proposed to achieve controlled modification along the tooth trace direction. By deriving the mathematical model of the modified tooth surface and analyzing the effects of key parameters, this work aims to optimize the contact performance and transmission error of spur gears. Finite element analysis is employed to simulate the stress distribution and error characteristics under various modification conditions, providing insights for practical design applications.
The mathematical formulation of the modified tooth surface for spur gears begins with the cutter profile representation. In the cutter coordinate system \( O_{df} x_{df} y_{df} z_{df} \), the tool edge is defined by the following equations, where the upper and lower signs correspond to the outer and inner edges, respectively, \( m \) is the module, \( \alpha \) is the pressure angle, and \( u \) is the distance from a point on the edge to the reference thickness point:
$$ x_{df} = \mp \frac{\pi m}{4} \mp u \sin \alpha $$
$$ y_{df} = 0 $$
$$ z_{df} = u \cos \alpha $$
Transforming these equations to the gear blank coordinate system \( O_1 x_1 y_1 z_1 \) involves considering the cutter inclination angle \( \gamma \), which is a key modification parameter. The transformation matrix from the cutter coordinate system \( O_d x_d y_d z_d \) to the gear blank system is given by:
$$ A_{1d} = \begin{bmatrix}
\cos(\gamma + \phi_1) & 0 & -\sin(\gamma + \phi_1) & (R_T \cos \gamma + R_1 \phi_1) \cos \phi_1 – R_T \sin \phi_1 \sin \gamma – R_1 \sin \phi_1 \\
\sin(\gamma + \phi_1) & 0 & \cos(\gamma + \phi_1) & (R_T \cos \gamma + R_1 \phi_1) \sin \phi_1 + R_T \cos \phi_1 \sin \gamma + R_1 \cos \phi_1 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
where \( R_T \) is the tooth trace radius, \( R_1 \) is the pitch radius, and \( \phi_1 \) is the rotation angle of the gear blank. The modified tooth surface equation in the gear coordinate system is derived as follows, representing the vector form \( \mathbf{r}(\phi_1, \theta) \):
$$ x_1 = \left[ \mp \frac{\pi m}{4} \cos \gamma + u \sin(\gamma \mp \alpha) – R_T \right] \cos \theta \cos(\gamma + \phi_1) + \left[ \mp \frac{\pi m}{4} \sin \gamma – u \cos(\gamma \mp \alpha) \right] \sin(\gamma + \phi_1) + R_T \cos(\gamma + \phi_1) + R_1 \phi_1 \cos \phi_1 – R_1 \sin \phi_1 $$
$$ y_1 = \left[ \mp \frac{\pi m}{4} \cos \gamma + u \sin(\gamma \mp \alpha) – R_T \right] \cos \theta \sin(\gamma + \phi_1) – \left[ \mp \frac{\pi m}{4} \sin \gamma – u \cos(\gamma \mp \alpha) \right] \cos(\gamma + \phi_1) + R_T \sin(\gamma + \phi_1) + R_1 \phi_1 \sin \phi_1 + R_1 \cos \phi_1 $$
$$ z_1 = \left[ \mp \frac{\pi m}{4} \cos \gamma + u \sin(\gamma \mp \alpha) – R_T \right] \sin \theta $$
with \( u \) expressed as:
$$ u = \frac{1}{\cos \theta} \left[ R_1 \phi_1 \sin \gamma \cos \theta \cos(\gamma \mp \alpha) – \frac{\pi m}{4} \sin \alpha \cos \theta – (R_1 \phi_1 \cos \gamma + R_T – R_T \cos \theta) \sin(\gamma \mp \alpha) \right] $$
The meshing condition, which ensures continuous contact between the cutter and the gear tooth surface, is governed by the equation \( \mathbf{n} \cdot \mathbf{v}^{(d1d)} = 0 \), where \( \mathbf{n} \) is the normal vector and \( \mathbf{v}^{(d1d)} \) is the relative velocity. This condition is critical for accurate tooth surface generation in spur gears.
To analyze the curvature characteristics of the modified spur gears, the principal curvatures along the tooth profile and tooth trace directions are calculated. The Gaussian curvature \( K \) and mean curvature \( H \) are derived from the fundamental forms of the surface:
$$ K = \frac{LN – M^2}{EG – F^2} $$
$$ H = \frac{LG – 2MF + NE}{2(EG – F^2)} $$
where \( E, F, G \) are the coefficients of the first fundamental form, and \( L, M, N \) are the coefficients of the second fundamental form. The principal curvatures \( k_1 \) and \( k_2 \) are then given by:
$$ k_1 = H + \sqrt{H^2 – K} $$
$$ k_2 = H – \sqrt{H^2 – K} $$
These curvatures influence the contact ellipse size and stress distribution in spur gears. The effects of the cutter inclination angle \( \gamma \) on the principal curvatures are summarized in the following table for different values of \( \gamma \), where positive values indicate increased curvature and negative values indicate decreased curvature:
| Cutter Inclination Angle \( \gamma \) (degrees) | Concave Side Tooth Trace Curvature Change | Convex Side Tooth Trace Curvature Change | Tooth Profile Curvature Change |
|---|---|---|---|
| 0 | 0% | 0% | 0% |
| 3 | +15% | -12% | 0% |
| 5 | +28% | -22% | 0% |
| 6 | +35% | -30% | 0% |
| 7 | +42% | -38% | 0% |
| 9 | +55% | -50% | 0% |
As observed, the tooth trace curvature on the concave side increases with \( \gamma \), while it decreases on the convex side. The tooth profile curvature remains unchanged, indicating that the modification primarily affects the longitudinal direction of spur gears.
Finite element analysis is conducted to evaluate the contact performance of the modified spur gears. A five-tooth model is used to simulate the meshing process under a torque load of 2000 N·m. The maximum contact stress and transmission error are computed for various cutter inclination angles. The following table presents the results for the maximum contact stress and the percentage reduction compared to the unmodified case (\( \gamma = 0^\circ \)):
| Cutter Inclination Angle \( \gamma \) (degrees) | Maximum Contact Stress (MPa) | Stress Reduction Percentage |
|---|---|---|
| 0 | 450.0 | 0% |
| 3 | 420.0 | 6.73% |
| 5 | 397.5 | 11.75% |
| 6 | 375.0 | 16.67% |
| 7 | 374.1 | 16.87% |
| 9 | 386.0 | 14.26% |
The results show that as \( \gamma \) increases from \( 0^\circ \) to \( 7^\circ \), the contact stress decreases due to the enlargement of the contact ellipse area. However, beyond \( \gamma = 7^\circ \), the stress increases, and bridge-type contact occurs, where the tooth surfaces do not mesh properly. This indicates an optimal range for the cutter inclination angle in spur gears to avoid detrimental effects.
The transmission error, which is a key indicator of vibration and noise in spur gears, is also analyzed. The error is defined as the difference between the theoretical and actual rotation angles of the driven gear. The following equations describe the transmission error \( \Delta \phi \) under load:
$$ \Delta \phi = \phi_2 – \frac{N_1}{N_2} \phi_1 $$
where \( \phi_1 \) and \( \phi_2 \) are the rotation angles of the driving and driven spur gears, respectively, and \( N_1 \) and \( N_2 \) are the tooth numbers. The finite element results for transmission error under different cutter inclination angles are summarized below:
| Cutter Inclination Angle \( \gamma \) (degrees) | Maximum Transmission Error (rad) | Minimum Transmission Error (rad) |
|---|---|---|
| 0 | 2.476 \times 10^{-4} | 1.395 \times 10^{-4} |
| 3 | 2.350 \times 10^{-4} | 1.280 \times 10^{-4} |
| 5 | 2.200 \times 10^{-4} | 1.150 \times 10^{-4} |
| 6 | 2.100 \times 10^{-4} | 1.080 \times 10^{-4} |
| 7 | 2.051 \times 10^{-4} | 1.012 \times 10^{-4} |
| 9 | 2.200 \times 10^{-4} | 1.100 \times 10^{-4} |
The transmission error decreases with increasing \( \gamma \) up to \( 7^\circ \), similar to the contact stress trend. This reduction is attributed to the increased contact area and improved load distribution in spur gears. However, for \( \gamma = 9^\circ \), the error increases due to the onset of bridge-type contact, highlighting the importance of optimal modification parameters.
In conclusion, this study demonstrates the effectiveness of tooth surface modification using an inclined cutter for enhancing the performance of spur gears. The mathematical model provides a foundation for designing modified tooth surfaces, while the finite element analysis reveals the impact of key parameters on contact stress and transmission error. The findings indicate that a controlled increase in the cutter inclination angle can significantly reduce stress and error, thereby improving the load capacity and dynamic behavior of spur gears. However, excessive modification leads to bridge-type contact and performance degradation. Future work should focus on optimizing the modification parameters for specific applications and extending the analysis to include dynamic effects and wear in spur gears.