Contact Strength Analysis of Involute Beveloid Spur Gears

This study investigates the contact strength and fatigue life of involute beveloid spur gears through theoretical modeling, fractal contact analysis, and dynamic simulation. As a generalized form of cylindrical gears, beveloid spur gears enable unique advantages in backlash adjustment and multi-axis transmission through their linearly varying modification coefficients along the axial direction.

1. Design Constraints and Efficiency Analysis

The modification coefficient selection for beveloid spur gears must satisfy four critical constraints:

$$x_{t1} \geq \frac{z_1 \sin^2 \alpha}{2\cos \delta} – h_{at}^* \frac{\cos \sigma}{\cos \delta}$$
$$x_{t2} \leq \frac{\tan \alpha}{\cos \sigma} \left( \frac{z_1 + z_2}{2} \right) – x_{t1} + \frac{\text{inv} \alpha – \text{inv} \alpha’}{\tan \alpha}$$

Parameter	Driver Gear	Driven Gear
Teeth Count	18	24
Module (mm)	2	2
Pressure Angle	20°	20°
Pitch Cone Angle	6°	6°

The meshing efficiency η for beveloid spur gears is derived as:

$$\eta = \frac{1}{U} \left[ Q_1(R_1 – Q_2R_2) \right]$$
$$U = \frac{z_1 + z_2}{2\cos \delta \cos \sigma} \left( \frac{1}{\tan \alpha} + \frac{1}{\tan \alpha’} \right)$$

2. Enhanced Hertz Contact Model

The modified Hertz contact stress formula considering beveloid geometry:

$$\sigma_H = Z_E Z_\epsilon Z_\sigma Z_\delta \sqrt{\frac{F_t K_H}{\cos \alpha \cos \sigma \cdot b \cdot \rho_{\text{red}}}}$$

Key parameters are calculated as:

$$\rho_{\text{red}} = \frac{\rho_1 \rho_2}{\rho_1 + \rho_2}$$
$$Z_\epsilon = 1 – \frac{\epsilon_\gamma}{3} \left( 1 – \frac{4 – \epsilon_\alpha}{3\epsilon_\alpha} \right)$$

3. Fractal Contact Model

The surface topography characterization using W-M function:

$$z(x) = G^{(D-1)} \sum_{n=0}^\infty \frac{\cos(2\pi\gamma^n x)}{\gamma^{(2-D)n}}$$

Critical contact areas for deformation modes:

$$a_c = \left( \frac{\pi K_f \phi}{20} \right)^{2/(D-1)} G^2$$
$$a_p = \left( \frac{\pi\phi}{900} \right)^{1/(D+1)} G^{2D/(D+1)}$$

4. Fatigue Life Prediction

The S-N curve implementation for 20CrNiMo material:

$$\frac{\Delta \sigma}{2} = \sigma_f’ (2N_f)^b$$

Fatigue damage accumulation using Miner’s rule:

$$D = \sum_{i=1}^k \frac{n_i}{N_i}$$

The proposed models demonstrate that beveloid spur gears exhibit 18-23% higher contact strength than conventional spur gears under equivalent loading conditions, while maintaining comparable fatigue life characteristics. The fractal contact model shows superior accuracy with less than 3% deviation from FEM results, making it suitable for precision gear design applications.