In my work on micro and special motors, I have focused extensively on the improvement of worm gear reducers used in automatic doors and barrier gates. Traditional systems employed AC reducers with braking devices, which were bulky, complex, and costly. Moreover, when backup batteries were needed, they required large capacity and inverters. After 2000, permanent magnet DC worm gear reducer motors gradually replaced these conventional designs. However, the typical motors derived from automotive wiper motors suffered from low efficiency and poor load capacity. Therefore, I undertook a systematic approach to improve the worm gear pair.
Force and Loss Analysis of the Worm Gear Pair
Before any modification, I conducted a detailed force and loss analysis of the worm gear transmission. The key forces acting on the worm and worm gear are summarized in the following table, where all symbols are defined in the subsequent equations.
| Component | Expression | Remarks |
|---|---|---|
| Worm tangential force \(F_{t1}\) (equals worm gear axial force \(F_{x2}\)) | \(F_{t1} = -F_{x2} = \dfrac{200 T_1}{d_1}\) | \(T_1\) is input torque (Nm), \(d_1\) in mm |
| Worm axial force \(F_{x1}\) (equals worm gear tangential force \(F_{t2}\)) | \(F_{x1} = -F_{t2} = -\dfrac{200 T_2}{d_2 + 2 x_2 m}\) | \(T_2\) is output torque, \(d_2\) worm gear pitch diameter, \(x_2\) profile shift coefficient, \(m\) module |
| Worm radial force \(F_{r1}\) (equals worm gear radial force \(F_{r2}\)) | \(F_{r1} = -F_{r2} = -F_{t2} \tan \alpha_x\) | \(\alpha_x\) is axial pressure angle |
| Normal force \(F_n\) | \(F_n = \dfrac{F_{x1}}{\cos\gamma \cos\alpha_n} \approx -\dfrac{F_{t2}}{\cos\gamma \cos\alpha_x} = -\dfrac{200 T_2}{d_2 \cos\gamma \cos\alpha_x}\) | \(\gamma\) lead angle, \(\alpha_n\) normal pressure angle |
| Worm gear efficiency (worm driving) \(\eta\) | \(\eta = \dfrac{\tan\gamma}{\tan(\gamma + \rho_v)}\) | \(\rho_v\) equivalent friction angle |
| Sliding velocity \(V_S\) | \(V_S = \dfrac{V_1}{\cos\gamma} = \dfrac{d_1 n_1}{19090 \cos\gamma} = \dfrac{m n_1}{19090 \sin\gamma}\) | \(n_1\) in rpm, \(m\) module |
| Friction force \(F\) | \(F = \mu F_n = \dfrac{200 \mu T_2}{d_2 \cos\gamma \cos\alpha_x}\) | \(\mu\) coefficient of friction |
| Friction power loss \(P\) | \(P = F V_S = \dfrac{200 \mu T_2 m n_1}{19090 d_2 \cos\gamma \sin\gamma \cos\alpha_x} = \dfrac{400 \mu T_2 m n_1}{19090 d_2 \sin 2\gamma \cos\alpha_x}\) | Loss in Watts |
The equations above reveal that the efficiency of the worm gear reducer is strongly influenced by the lead angle \(\gamma\), the pressure angle \(\alpha_x\) (or \(\alpha_n\)), and the friction coefficient \(\mu\). The friction loss is inversely proportional to \(\sin 2\gamma\), which means that increasing the lead angle reduces loss, but practical limits exist due to manufacturing constraints. The normal force \(F_n\) is also reduced when \(\gamma\) is increased and \(\alpha_x\) is decreased, leading to lower frictional forces and higher efficiency.
Another critical aspect is the stiffness of the worm shaft. The deflection of the worm shaft under load affects the meshing quality and noise. For a simply supported shaft (both ends supported), the deflection \(y_1\) is given by:
$$y_1 = \frac{\sqrt{F_{t1}^2 + F_{r1}^2} \, L^3}{48 E I}$$
For a cantilevered shaft (one end supported), the deflection is twice as large:
$$y_1 = \frac{\sqrt{F_{t1}^2 + F_{r1}^2} \, L^3}{24 E I}$$
where \(L\) is the bearing span, \(E\) is Young’s modulus, and \(I\) is the area moment of inertia of the worm shaft. Reducing the radial force \(F_{r1}\) directly improves the shaft stiffness. From the radial force expression, \(F_{r1} = -F_{t2} \tan \alpha_x\), it is clear that a smaller pressure angle \(\alpha_x\) reduces the radial force, which not only lowers bending but also decreases bearing loads.
Design Improvements for Higher Efficiency
Based on the above analysis, I implemented several key modifications to the worm gear pair:
- Reduce the worm characteristic coefficient \(q\) (where \(q = d_1 / m\)). A smaller \(q\) means a smaller worm diameter for a given module, which allows a larger lead angle \(\gamma\). However, reducing \(q\) also reduces the worm root diameter and thus weakens the shaft. To compensate, I used a reduced tooth height (short tooth depth) or a dual-modulus design for the worm and worm gear geometry. This approach increases the root circle diameter while maintaining the same center distance, thereby improving the worm shaft’s bending strength.
- Increase the lead angle \(\gamma\). A larger lead angle directly improves efficiency as seen from the efficiency formula. The lead angle is related to the number of starts \(Z_1\) and the axial module \(m_x\): \(\tan \gamma = Z_1 m_x / d_1\). By increasing \(Z_1\) or decreasing \(d_1\), I achieved a higher \(\gamma\).
- Decrease the pressure angle \(\alpha_n\). I reduced the normal pressure angle from the typical 20° to 8° (in the application example). This reduction not only lowers the radial force but also decreases the normal force, as shown in the normal force equation. However, a lower pressure angle may reduce tooth strength under static loads, so careful checking of tooth bending and contact stress is required. In my application, the reduced pressure angle proved satisfactory for the low-speed, high-torque duty cycle of automatic doors.
- Support the worm shaft at both ends. Instead of the common cantilever support (often used in wiper motors), I designed the housing to accommodate bearings on both sides of the worm. This halves the deflection at the meshing point, significantly improving the contact pattern and reducing wear. The stiffness improvement is evident from the deflection formulas: the two-support configuration gives one-quarter of the deflection of a cantilever with the same length.
Application Example and Parameter Comparison
A practical implementation of these improvements is shown in the following table, which lists the worm parameters used in a redesigned reducer. The tooth profile was also modified to ensure proper clearance and lubrication.
| Parameter | Symbol | Value |
|---|---|---|
| Number of starts (threads) | \(Z_1\) | 1 |
| Axial module | \(m_x\) | 1.25 mm |
| Lead angle (direction) | \(\lambda\) | 9.7824° (right hand) |
| Axial pressure angle | \(\alpha_n\) | 8° |
The tooth shape used is depicted above. Compared to the conventional motor, the efficiency improvement reached 6% under typical operating conditions. Additional adjustments to the worm tooth thickness were made based on wear tests during development, ensuring optimal matching with the worm gear over the product life.
Further Considerations: Gap Adjustment and Blank Dimension
In the assembly of the worm gear reducer, I also addressed the clearance between inner and outer sliding blocks (or similar components) which should equal the thickness of one layer of material. By adjusting the diameter of guide pulleys and the thickness of guide plates, the gap can be tuned. Alternatively, an adjustable wedge can be used for more flexible clearance control.
Moreover, for components that are formed from a cylindrical or drum-shaped blank (sometimes used in worm gear housings or output shafts), the blank dimensions must be calculated correctly. For a drum-shaped blank (Figure 5 in the original text but not referenced here by number), the dimension \(a\) before forming is given by:
$$a = 2\sqrt{(R + t)^2 – \left(\frac{H}{2}\right)^2} – 2t$$
where:
- \(a\) — dimension before forming (mm)
- \(R\) — drum radius (mm)
- \(H\) — workpiece height (mm)
- \(t\) — workpiece thickness (mm)
This formula ensures that after forming the workpiece fits the required geometry exactly. Although not directly related to the worm gear itself, this calculation is relevant for manufacturing jigs and fixtures used in reducer assembly.
Conclusion
Through a comprehensive force and loss analysis, I identified that the efficiency of a worm gear reducer can be significantly improved by reducing the worm characteristic coefficient, increasing the lead angle, decreasing the pressure angle, and supporting the worm shaft at both ends. The friction loss formula shows that a larger lead angle and smaller pressure angle directly reduce power dissipation. The stiffness analysis confirms that a smaller pressure angle and two-end support reduce the worm deflection, enhancing meshing quality and reducing wear. Application of these principles to a permanent magnet DC motor resulted in a 6% efficiency gain, demonstrating the effectiveness of the approach. Future work may involve optimizing the tooth profile further and exploring advanced materials for the worm gear to reduce friction coefficient \(\mu\). The improvements described here are readily applicable to other worm gear reducers in servo systems, industrial automation, and automotive auxiliary drives.