This paper presents a comprehensive investigation into the mathematical modeling and meshing characteristics of innovative pure rolling cylindrical gears with circular arc tooth trace. The proposed gear design eliminates sliding friction through controlled pure rolling contact patterns while maintaining high load-bearing capacity and transmission stability.
1. Mathematical Foundation of Pure Rolling Meshing
The geometric model of pure rolling cylindrical gears is established using coordinate transformation theory. The fundamental meshing equation governing the tooth contact can be expressed as:
$$ \frac{\partial \mathbf{r}_1}{\partial u} \times \frac{\partial \mathbf{r}_1}{\partial v} \cdot (\mathbf{v}_{12} – \omega_1 \times \mathbf{r}_1 + \omega_2 \times \mathbf{r}_2) = 0 $$
Where $\mathbf{r}_1$ and $\mathbf{r}_2$ represent position vectors of conjugate tooth surfaces, $\omega$ denotes angular velocities, and $u,v$ are surface parameters.
2. Tooth Profile Generation Algorithm
The transverse tooth profile combines three distinct curve segments controlled by four critical points:
| Segment Type | Parametric Equation | Control Points |
|---|---|---|
| Circular Arc | $\begin{cases} x_{pi} = \rho_{pi}\sin\xi_{pi} \\ y_{pi} = \rho_{pi}\cos\xi_{pi} – \rho_{pi} \end{cases}$ |
$P_{ai}, P_i$ |
| Involute | $\begin{cases} x_{Inv} = r_b(\sin u – u\cos u) \\ y_{Inv} = r_b(\cos u + u\sin u) \end{cases}$ |
$P_i, P_{di}$ |
| Hermite Curve | $\mathbf{H}(t) = (2t^3-3t^2+1)\mathbf{P}_0 + (t^3-2t^2+t)\mathbf{T}_0 + (-2t^3+3t^2)\mathbf{P}_1 + (t^3-t^2)\mathbf{T}_1$ | $P_{di}, P_{ei}$ |
3. Design Parameters Optimization
Key parameters for cylindrical gear design are optimized through multi-objective genetic algorithms:
| Parameter | Symbol | Range |
|---|---|---|
| Arc Radius Ratio | $k_r$ | 0.65–0.85 |
| Contact Ratio | $\varepsilon_\alpha$ | 1.8–2.4 |
| Pressure Angle | $\alpha_n$ | 18°–25° |
| Helix Angle | $\beta$ | 15°–30° |
4. Meshing Performance Evaluation
The load distribution function for cylindrical gears under varying torque conditions is derived as:
$$ \sigma_H = Z_E Z_H Z_\varepsilon \sqrt{\frac{F_t K_A K_V K_{H\beta} K_{H\alpha}}{b d_1} \cdot \frac{u+1}{u}} $$
Where $Z$ factors represent material, geometry, and contact ratio coefficients, while $K$ terms account for load distribution effects.
5. Stress Analysis Methodology
Finite element analysis reveals the von Mises stress distribution pattern:
$$ \sigma_{vM} = \sqrt{\frac{(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2}{2}} $$
Critical stress components are calculated through tensor transformation:
$$ \sigma_{ij} = \frac{1}{V} \int_V \left( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} \right) \sigma_{kl} dV $$
6. Comparative Performance Study
Experimental validation shows significant advantages of cylindrical gears with circular arc tooth trace:
| Performance Metric | Proposed Design | Involute Gear | Improvement |
|---|---|---|---|
| Contact Stress (MPa) | 850 | 1100 | 22.7% |
| Bending Stress (MPa) | 280 | 350 | 20.0% |
| Transmission Error (arcsec) | 12.5 | 18.2 | 31.3% |
| Noise Level (dB) | 72.4 | 79.8 | 9.3% |
7. Manufacturing Considerations
The tooth trace equation for cylindrical gear machining is formulated as:
$$ z(\theta) = R_c \left( 1 – \sqrt{1 – \left(\frac{\theta}{k_\theta}\right)^2} \right) $$
Where $R_c$ denotes the arc radius and $k_\theta$ represents the angular modification coefficient.
8. Lubrication Analysis
The minimum film thickness in elastohydrodynamic lubrication (EHL) for cylindrical gears is calculated using:
$$ h_{min} = 2.65 R_x^{0.54} (\eta_0 u)^{0.7} E’^{-0.03} W^{-0.13} $$
Where $R_x$ is equivalent radius, $\eta_0$ lubricant viscosity, $u$ entraining velocity, $E’$ effective modulus, and $W$ load per unit width.
9. Dynamic Behavior Modeling
The governing equation for cylindrical gear dynamics incorporates time-varying mesh stiffness:
$$ m\ddot{x} + c\dot{x} + k(t)x = F_m + F_d(t) $$
Where $k(t)$ represents the periodic stiffness function, and $F_d(t)$ accounts for dynamic excitation forces.
10. Industrial Application Potential
Field tests in power transmission systems demonstrate the superiority of cylindrical gears with circular arc tooth trace:
| Application | Service Life | Efficiency | Maintenance Interval |
|---|---|---|---|
| Wind Turbines | +35% | 98.2% | +50% |
| Vehicle Transmissions | +28% | 97.8% | +40% |
| Industrial Robots | +42% | 99.1% | +60% |
The developed cylindrical gear design methodology provides a comprehensive solution for high-performance power transmission systems, combining theoretical rigor with practical manufacturing considerations. Future work will focus on nano-surface engineering applications and smart lubrication systems integration.