Gear transmission is a critical method for power transmission, and researchers have conducted extensive studies on various gear types, including spur gears, to enhance performance and reliability. In this study, we focus on a novel cylindrical gear known as the Variable Hyperbolic Circular-Arc-Tooth-Trace (VH-CATT) cylindrical gear, which evolved from the Gleason spiral bevel gear. This gear type offers high load-carrying capacity, transmission efficiency, low installation accuracy requirements, and no axial force. Our research explores a tooth surface modification method for VH-CATT cylindrical gears using an inclined cutter head during the milling process, and we analyze the dynamic meshing characteristics under different modification parameters. First, we derive the modified tooth surface equation based on the forming principle of VH-CATT cylindrical gears and reconstruct the tooth surface. Second, we establish kinematic analysis models in Adams for VH-CATT cylindrical gears with varying modification parameters. Finally, we perform kinematic analyses on the modified gear pairs to determine the driven wheel’s speed fluctuations and the evolution of dynamic meshing forces. The results indicate that as the modification parameter increases to a certain value, the average speed of the driven wheel remains relatively stable, but the fluctuation amplitude decreases, along with a reduction in the average meshing force. Beyond this point, further increases lead to greater speed fluctuations and higher meshing forces. This study provides a theoretical foundation for vibration reduction, noise control, and modification design in gear systems, with particular relevance to spur gears and their applications.
The VH-CATT cylindrical gear is manufactured using a large cutter head with dual-edge milling, and its forming principle is detailed in prior literature. To improve dynamic meshing characteristics and reduce vibration and noise, we propose a modification method involving an inclined cutter head during machining. The mathematical model for the modified tooth surface is derived from the gear engagement theory and coordinate systems of the machining process. The tooth surface equation is expressed as:
$$ r(u, \theta) = \begin{bmatrix} (R_T \pm u \sin \alpha) \cos \theta + (R_T \pm u \sin \alpha) \sin \theta \cdot \tan \beta \\ -(R_T \pm u \sin \alpha) \sin \theta + (R_T \pm u \sin \alpha) \cos \theta \cdot \tan \beta \\ u \cos \alpha \end{bmatrix} $$
where the upper sign corresponds to the outer blade and the lower sign to the inner blade, $\alpha$ is the tool pressure angle, $m$ is the module, $R_T$ is the nominal tooth trace radius, $u$ is the distance from any point on the blade surface to the point of equal tooth thickness, $\theta$ is the tool rotation angle from the gear blank’s mid-section to the end face, $\beta$ is the cutter head inclination angle, and $\phi_1$ is the gear blank rotation angle. This equation allows for precise reconstruction of the tooth surface, accounting for the modification introduced by the inclined cutter head.
For three-dimensional modeling, we use MATLAB for numerical computation of the concave and convex tooth surfaces of a single gear tooth, followed by importation into UG software for solid modeling. The gear pair parameters are set as follows: number of teeth $z_1 = 29$ and $z_2 = 41$, module $m = 8 \, \text{mm}$, pressure angle $\alpha = 20^\circ$, face width $b = 60 \, \text{mm}$, and cutter head radius $R_T = 180 \, \text{mm}$. By varying the cutter head inclination angle $\beta$, we generate accurate 3D models of the modified VH-CATT cylindrical gears. The modeling process ensures that the tooth geometry reflects the modification effects, which is crucial for subsequent dynamic analysis.
To investigate the dynamic meshing performance, we establish virtual models in Adams for kinematic and dynamic simulations. The models include material properties, motion pairs, and contact forces to simulate real-world operating conditions. For gear pair meshing, the contact force stiffness coefficient is determined based on Hertzian contact theory. The stiffness coefficient $k$ is calculated as:
$$ k = \frac{4}{3} E^* \sqrt{R^*} $$
where $E^*$ is the equivalent Young’s modulus and $R^*$ is the equivalent radius of curvature at the contact point. The equivalent radius $R^*$ is given by:
$$ \frac{1}{R^*} = \frac{1}{R_1} + \frac{1}{R_2} $$
and the equivalent Young’s modulus $E^*$ is:
$$ \frac{1}{E^*} = \frac{1 – \nu_1^2}{E_1} + \frac{1 – \nu_2^2}{E_2} $$
Here, $E_1 = E_2 = 2.06 \times 10^{11} \, \text{Pa}$ and $\nu_1 = \nu_2 = 0.3$ for the gear materials. A load of $1500 \, \text{N} \cdot \text{m}$ is applied to the driven wheel to simulate operational conditions. The simulation parameters are set to analyze the effects of different cutter head inclination angles $\beta$ (e.g., $0^\circ$, $3^\circ$, $5^\circ$, $7^\circ$) on speed fluctuations and meshing forces, with a focus on comparisons to spur gears in terms of dynamic behavior.
The impact of modification parameters on speed fluctuations is analyzed by examining the driven wheel’s rotational speed under various $\beta$ values. The results show periodic speed variations with fluctuations around a mean value, indicating stable transmission. The table below summarizes the maximum, minimum, and average speeds, along with speed deviation percentages for different $\beta$:
| Cutter Head Inclination Angle $\beta$ (°) | Maximum Speed (deg/s) | Minimum Speed (deg/s) | Average Speed (deg/s) | Speed Deviation (%) |
|---|---|---|---|---|
| 0 | 2657.71 | 2224.44 | 2476.28 | 0.027 |
| 3 | 2654.44 | 2225.73 | 2475.19 | 0.017 |
| 5 | 2656.56 | 2229.99 | 2475.89 | -0.011 |
| 7 | 2658.51 | 2241.21 | 2475.78 | -0.007 |
The data reveals that changes in $\beta$ have minimal impact on the average speed but affect the fluctuation amplitude. As $\beta$ increases to around $5^\circ$, the speed deviation decreases, suggesting improved transmission stability. However, beyond this point, further increases in $\beta$ lead to higher deviations, indicating a reduction in stability. This behavior highlights the importance of optimal modification parameters for enhancing performance, similar to considerations in spur gears design.
Next, we analyze the dynamic meshing forces under different modification parameters. The meshing force is critical for understanding impact characteristics and vibration in gear systems. The table below presents the maximum, minimum, and average meshing forces for $\beta$ values ranging from $0^\circ$ to $7^\circ$:
| Cutter Head Inclination Angle $\beta$ (°) | Maximum Meshing Force (N) | Minimum Meshing Force (N) | Average Meshing Force (N) |
|---|---|---|---|
| 0 | 30866.0 | 3258.5 | 17062.3 |
| 1 | 30304.5 | 3471.8 | 16888.2 |
| 2 | 30550.8 | 2245.0 | 16397.9 |
| 3 | 30143.9 | 268.0 | 15606.0 |
| 4 | 29942.2 | 562.0 | 15252.1 |
| 5 | 29829.3 | 356.8 | 15093.1 |
| 6 | 29261.3 | 289.1 | 14775.2 |
| 7 | 30913.9 | 1860.6 | 16387.3 |
The average meshing force decreases as $\beta$ increases up to $6^\circ$, reaching a minimum of $14775.2 \, \text{N}$, but then rises at $\beta = 7^\circ$. This trend demonstrates that moderate modification reduces meshing forces, contributing to lower vibration and noise, which is a key advantage in spur gears applications. The relationship between $\beta$ and average meshing force can be approximated by a quadratic function:
$$ F_{\text{avg}} = a \beta^2 + b \beta + c $$
where $a$, $b$, and $c$ are constants derived from regression analysis. For instance, based on the data, $F_{\text{avg}}$ minimizes around $\beta = 6^\circ$, emphasizing the need for precise modification control.
In conclusion, our study on VH-CATT cylindrical gears demonstrates that tooth surface modification via an inclined cutter head significantly influences dynamic meshing characteristics. The speed fluctuations and meshing forces are optimized at specific cutter head inclination angles, with improvements in stability and force reduction up to a threshold. These findings align with principles in spur gears design, where modification techniques are employed to enhance performance. Future work could explore additional parameters, such as load variations and thermal effects, to further refine gear transmission systems. This research underpins the development of efficient and quiet gear drives, benefiting industries reliant on high-performance spur gears and similar components.