In the field of advanced mechanical transmissions, the design and analysis of non-orthogonal asymmetric helical face gears have garnered significant attention due to their unique geometric properties and applications in high-precision equipment. These gears feature variable shaft angles and differing pressure angles on both sides of the tooth, which enhance their adaptability and performance in complex transmission systems. This article delves into the tooth surface design and meshing characteristics of non-orthogonal asymmetric helical face gears, employing mathematical modeling, improved potential energy methods, and finite element simulations to compute time-varying meshing stiffness under various pressure angles and shaft angles. The findings provide crucial insights for optimizing the reliability and efficiency of helical gear transmission systems.
The tooth surface of a non-orthogonal asymmetric helical face gear is generated by enveloping the tooth surface of a hob cutter. To derive the gear’s tooth surface, a series of coordinate systems are established to represent the relative motion between the gear and the cutter. The primary coordinate systems include the rotating coordinate system of the helical gear, the fixed coordinate system of the face gear, and auxiliary systems for transformation. The transformation matrices between these systems are essential for accurately describing the gear geometry. For instance, the transformation from the hob cutter’s fixed coordinate system to its rotating system is given by:
$$ M_{p,2} = \begin{bmatrix} \cos\omega_1 & -\sin\omega_1 & 0 & 0 \\ \sin\omega_1 & \cos\omega_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Similarly, the transformation to the face gear’s rotating system involves a shaft angle γ, expressed as:
$$ M_{2,1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\gamma & \sin\gamma & 0 \\ 0 & -\sin\gamma & \cos\gamma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The overall transformation matrix from the hob cutter to the face gear coordinate system combines these matrices, facilitating the derivation of the tooth surface equations. The position vectors and normal vectors for the drive and coast sides of the asymmetric helical gear hob are defined based on base circle radii, pressure angles, and helical parameters. For example, the drive side position vector is:
$$ \mathbf{r}_d(\theta_d, e_d) = \begin{bmatrix} r_{bd} (\sin\kappa_d – \theta_d \cos\kappa_d) \\ -r_{bd} (\cos\kappa_d + \theta_d \sin\kappa_d) \\ \frac{L_l e_d}{2\pi} \\ 1 \end{bmatrix} $$
where κ_d is a function of pressure angle and tooth parameters, and L_l is the lead of the helix. The meshing equations for the drive and coast sides are derived to ensure proper contact conditions. For the drive side, the meshing equation is:
$$ f_d(e_d, \theta_d, \omega_1) = r_{bd} \cos\beta \left(1 – \frac{N_1}{N_2} \cos\gamma\right) – \frac{r_{bd} N_1}{N_2} \sin\beta \sin\gamma \left[ \sin\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right) – \theta_d \cos\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right) \right] – \frac{N_1}{N_2} \left[ \sin\gamma \sin\beta + \cos\gamma \cos\beta \sin\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right) \right] – \frac{N_1}{N_2} e_d \sin\gamma \cos\beta \cos\left(\omega_1 + \frac{\pi}{2N_1} – \tan\alpha_d – \alpha_d\right) $$
These equations are solved to obtain the tooth surface coordinates, which are then discretized and modeled using CAD software to generate the complete helical gear surface. The design parameters for a typical non-orthogonal asymmetric helical face gear transmission system are summarized in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| Shaft Angle | γ | 60° |
| Helix Angle | β | 20° |
| Drive Side Pressure Angle | α_d | 30° |
| Coast Side Pressure Angle | α_c | 25° |
| Number of Hob Teeth | N_1 | 48 |
| Number of Face Gear Teeth | N_2 | 125 |
| Number of Helical Gear Teeth | N_3 | 45 |
| Module | m | 3 mm |
| Minimum Inner Radius | R_1 | 182 mm |
| Maximum Outer Radius | R_2 | 215 mm |
To avoid tooth root undercutting and tip sharpening, constraints on the tooth width are imposed. The conditions for singular points on the tooth surface are analyzed to determine the minimum inner radius and maximum outer radius. For instance, the radius for the drive side singular point is calculated as R_d1 = √(x_dgq² + y_dgq²), and the overall minimum inner radius is the maximum of the drive and coast side values. Similarly, the maximum outer radius is derived from points where tooth thickness approaches zero.
The time-varying meshing stiffness of the helical gear is computed using an improved potential energy method combined with a slicing technique. The gear tooth is divided into multiple trapezoidal slices along the axis, and the stiffness of each slice is calculated independently. The area moment of inertia for the i-th slice is given by:
$$ I_i = \frac{h_i^3 (a_i^2 + 4a_i b_i + b_i^2)}{36(a_i + b_i)} $$
where h_i is the height, and a_i and b_i are the lengths of the top and bottom bases of the slice. The bending stiffness k_w of the helical gear tooth is derived by integrating the deformations due to bending and shear across the slices. The formula involves summing contributions from all slices, considering the load distribution and geometric parameters:
$$ k_w = \int_{y_f}^{y_f + d} \int_{x_{t1}}^{x_{t2}} \frac{F \cos\alpha_i \, dx_a \, dy_a}{N} $$
where N represents the total deflection from bending and shear, expressed as a sum over slices. The base coupling stiffness k_b for single-tooth meshing is calculated using empirical relations based on Hertzian contact theory:
$$ \frac{1}{k_b} = \frac{\cos^2\alpha_i C_1 (V_1 \tan^2\alpha_i + 1)}{I_i E d} + \frac{\cos^2\alpha_i S B_1 L_f (1 – \nu) + G_1 L_f^2}{S^2 I_i E d} $$
Here, C_1, V_1, B_1, G_1 are polynomial fitting parameters, L_f is the distance from the contact point to the base, and ν is Poisson’s ratio. For double-tooth meshing, the base stiffness accounts for interactions between tooth pairs, with separate expressions for preceding and following teeth. The contact stiffness k_u is approximated as:
$$ \frac{1}{k_u} = \frac{1.37}{F^{0.1} E^{0.9} d^{0.8}} $$
The overall meshing stiffness for single-tooth and double-tooth engagement are then:
$$ k_{m1} = \frac{1}{\frac{1}{k_w} + \frac{1}{k_b} + \frac{1}{k_u}} $$
$$ k_{m2} = \frac{1}{\frac{1}{k_w} + \frac{1}{k_{b1}} + \frac{1}{k_{b2}} + \frac{1}{k_u}} $$
Analyzing the influence of pressure angle on the helical gear’s time-varying meshing stiffness reveals that as the pressure angle increases from 15° to 30° with a fixed shaft angle of 60°, the contact path shifts toward the outer diameter, and the meshing stiffness significantly increases. This enhancement improves the transmission system’s stability and load capacity. Conversely, when the pressure angle is fixed at 20° and the shaft angle varies from 30° to 90°, the meshing stiffness decreases with increasing shaft angle. Reducing the shaft angle from 90° to 30° boosts the stiffness, which is beneficial for system performance. The following table summarizes the average meshing stiffness values under different conditions, demonstrating these trends.
| Condition | Pressure Angle (°) | Shaft Angle (°) | Average Stiffness (N/mm) |
|---|---|---|---|
| Fixed Shaft Angle | 15 | 60 | 4.2 × 10⁴ |
| Fixed Shaft Angle | 20 | 60 | 5.1 × 10⁴ |
| Fixed Shaft Angle | 25 | 60 | 6.3 × 10⁴ |
| Fixed Shaft Angle | 30 | 60 | 7.5 × 10⁴ |
| Fixed Pressure Angle | 20 | 30 | 7.8 × 10⁴ |
| Fixed Pressure Angle | 20 | 45 | 6.9 × 10⁴ |
| Fixed Pressure Angle | 20 | 60 | 5.1 × 10⁴ |
| Fixed Pressure Angle | 20 | 90 | 4.0 × 10⁴ |
Finite element simulations are conducted to validate the computed meshing stiffness values. The tooth surface point cloud, derived from the meshing equations, is imported into modeling software to create a 3D model of the helical gear. After meshing, the model is analyzed under defined material properties and boundary conditions. The helical gear is set as the driving component, and the rotational degrees of freedom are constrained to simulate meshing. The output meshing force and rotation angles are monitored to derive the time-varying stiffness. The simulation results align closely with the theoretical calculations, with errors within 5%, confirming the accuracy of the proposed method. This validation underscores the reliability of the design and analysis approach for non-orthogonal asymmetric helical face gears.
In conclusion, the design and meshing stiffness analysis of non-orthogonal asymmetric helical face gears provide valuable insights for enhancing transmission system performance. The use of coordinate transformations and meshing equations enables precise tooth surface generation, while the improved potential energy method offers an efficient means to compute time-varying stiffness. The study demonstrates that increasing the pressure angle or decreasing the shaft angle can significantly improve the meshing stiffness of helical gears, thereby boosting system reliability. These findings contribute to the optimization of helical gear transmissions in demanding applications, ensuring higher efficiency and durability. Future work may focus on dynamic behavior analysis and experimental validation to further refine the design principles.