This paper presents a novel precision worm gear reducer featuring triple-worm drive mechanism for industrial robotics applications. Through mathematical modeling, structural optimization, and experimental validation, the study establishes a comprehensive design framework addressing critical performance parameters including transmission accuracy, backlash, and efficiency.
1. Mathematical Modeling of Triple-Worm Drive System
The coordinate transformation matrix for worm gear engagement is derived as:
$$B = \begin{bmatrix}
\cos\phi & -\sin\phi & 0 & 0 \\
\sin\phi & \cos\phi & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}$$
The relative velocity vector between worm and gear is expressed as:
$$v^{(12)} = \omega_1 \times r_1 – \omega_2 \times r_2$$
Where angular velocities satisfy:
$$\frac{\omega_1}{\omega_2} = \frac{z_2}{z_1}$$
| Parameter | Worm | Worm Gear |
|---|---|---|
| Module (mm) | 1.5 | 1.5 |
| Number of Teeth | 1 | 45 |
| Pressure Angle | 20° | 20° |
| Material | 40Cr | ZCuSn10P1 |
2. Structural Design and Optimization
The three-stage transmission system achieves total reduction ratio:
$$i_{total} = i_1 \times i_2 \times i_3 = 2.4 \times 3 \times 45 = 324$$
Critical design equations for gear parameters:
$$d = \frac{mz}{\cos\beta}$$
$$\varepsilon_\alpha = \frac{1}{2\pi}\left[z_1(\tan\alpha_{a1} – \tan\alpha_w) + z_2(\tan\alpha_{a2} – \tan\alpha_w)\right]$$
| Parameter | Theoretical | FEA |
|---|---|---|
| Max Stress (MPa) | 300.78 | 326.97 |
| Axial Stiffness (N/m) | 1×10⁸ | 2.5×10⁷ |
| Natural Frequency (Hz) | – | 4165.3 |
3. Dynamic Performance Analysis
The transmission error spectrum reveals characteristic frequencies:
$$f_{mesh} = \frac{nz}{60} = \frac{1400 \times 45}{60} = 1050\ \text{Hz}$$
Load distribution coefficient calculation:
$$B_{sp} = \frac{nF_{sp}^{\text{max}}}{\sum F_{sp}}$$
| Center Distance (mm) | 41 | 44 | 47 |
|---|---|---|---|
| Load Coefficient | 1.016 | 1.023 | 1.030 |
4. Experimental Validation
Prototype testing results demonstrate:
$$\Delta \theta_{\text{max}} = 23.2^{\prime\prime}$$
$$\eta = \frac{T_{\text{out}}}{T_{\text{in}}i} \times 100\% = 76.97\%$$
| Speed (rpm) | 200 | 600 |
|---|---|---|
| 200Nm | 73.34% | 68.96% |
| 600Nm | 80.65% | 76.97% |
The developed worm gear reducer achieves remarkable performance metrics:
- Transmission accuracy: 40.13 arcsec
- Backlash: ≤0.47 arcmin
- Service life: >6000 hours
This research provides a complete technical solution for precision worm gear reducers in industrial robotics, demonstrating significant improvements in load capacity and motion accuracy compared to conventional designs. Future work will focus on lubrication optimization and mass production quality control.