In modern mechanical transmission systems, worm gear drives are widely used due to their high reduction ratio, compact structure, and smooth operation. However, the worm wheel ring is often made of precious non‑ferrous metals such as bronze to reduce friction and wear. To lower material cost, it is crucial to minimize the volume of the worm wheel ring while satisfying all strength and rigidity constraints. This paper presents an optimization design approach for worm gear transmission, taking the volume of the worm wheel ring as the objective. The design variables include the number of worm threads, module, and diameter coefficient. Ten constraints are formulated, covering geometric limits, contact strength, and worm shaft deflection. A numerical example demonstrates that the optimized design reduces the ring volume by 16.5% compared with the conventional design.
Introduction
Worm gear transmissions are essential in many industrial applications where high torque and low speed are required. The worm wheel is typically manufactured in a composite structure: a steel hub with a bronze ring. Since bronze is expensive, engineers strive to reduce the ring volume without compromising performance. Traditional design methods rely on empirical rules and iterative checks, which often yield over‑conservative dimensions. Optimization techniques provide a systematic way to find the best combination of design parameters. In this work, we focus on a worm gear drive with a fixed transmission ratio, and we aim to minimize the volume of the worm wheel ring. The key parameters influencing the ring volume are the number of worm threads z1, the module m, and the diameter coefficient q. The optimization problem is a nonlinear constrained minimization with three design variables and ten constraints.
The worm gear drive under consideration is enclosed, with the worm shaft supported by bearings. The worm wheel is a hobbed gear with a bronze rim attached to a cast‑iron or steel hub. The ring dimensions are determined by the geometry of the worm gear pair. By formulating the ring volume as a function of z1, m, and q, we can apply penalty function methods to find the optimum.
Mathematical Model
Objective Function
Referring to the structure of the worm wheel ring (Fig. 1 in the original paper), the ring volume V is given by:
$$
V = \frac{\pi b}{4} \left( d_e^2 – d_0^2 \right)
$$
where b is the face width of the worm wheel, de is the outside diameter of the ring, and d0 is the inner diameter. These dimensions are expressed in terms of the worm gear parameters:
$$
\begin{aligned}
d_e &= d_a + \frac{6m}{z_1+2} = m z_2 + 2m + \frac{6m}{z_1+2} \\
d_0 &= d_f – 2m = m z_2 – 4.4 m
\end{aligned}
$$
Here z2 = u z1 is the number of worm wheel teeth, u is the gear ratio, and φ is the face width coefficient (φ = 0.75 for z1 = 1–2; φ = 0.67 for z1 = 3–4). Substituting these expressions into the volume formula yields:
$$
V = \frac{\pi}{4} \phi m (q+2) \left[ \left( m u z_1 + 2m + \frac{6m}{z_1+2} \right)^2 – \left( m u z_1 – 4.4 m \right)^2 \right]
$$
After simplifying, the objective function becomes:
$$
f(\mathbf{X}) = \frac{\pi \phi m^3 (q+2)}{4} \left[ \left( u z_1 + 2 + \frac{6}{z_1+2} \right)^2 – \left( u z_1 – 4.4 \right)^2 \right]
$$
Since the gear ratio u is a known constant, we choose three design variables:
$$
\mathbf{X} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} z_1 \\ m \\ q \end{bmatrix}
$$
Thus the objective function in compact form is:
$$
f(\mathbf{X}) = \frac{\pi \phi x_2^3 (x_3+2)}{4} \left[ \left( u x_1 + 2 + \frac{6}{x_1+2} \right)^2 – \left( u x_1 – 4.4 \right)^2 \right]
$$
Design Variables and Constraints
Ten constraints are imposed to ensure the worm gear drive meets all practical requirements. They are grouped into geometric limits, strength conditions, and rigidity. The constraints are formulated as inequality functions gi(X) ≤ 0.
1. Constraints on worm thread number
For power transmission, the number of worm threads is usually between 2 and 4:
$$
\begin{aligned}
g_1(\mathbf{X}) &= 4 – x_1 \le 0 \\
g_2(\mathbf{X}) &= x_1 – 2 \le 0
\end{aligned}
$$
2. Constraints on worm wheel teeth number
The worm wheel tooth count is recommended to be between 30 and 80:
$$
\begin{aligned}
g_3(\mathbf{X}) &= 80 – u x_1 \le 0 \\
g_4(\mathbf{X}) &= u x_1 – 30 \le 0
\end{aligned}
$$
3. Constraints on module
The module should be between 2 and 18 mm for typical power drives:
$$
\begin{aligned}
g_5(\mathbf{X}) &= 18 – x_2 \le 0 \\
g_6(\mathbf{X}) &= x_2 – 2 \le 0
\end{aligned}
$$
4. Constraints on diameter coefficient
The diameter coefficient q usually ranges from 8 to 16:
$$
\begin{aligned}
g_7(\mathbf{X}) &= 16 – x_3 \le 0 \\
g_8(\mathbf{X}) &= x_3 – 8 \le 0
\end{aligned}
$$
5. Contact strength constraint
The worm wheel tooth flank must withstand the Hertzian contact stress. The condition is:
$$
m^3 q \ge K T_2 \left( \frac{500}{z_2 [\sigma_H]} \right)^2
$$
where K is the load factor, T2 is the output torque, and [σH] is the permissible contact stress. Rearranging gives:
$$
g_9(\mathbf{X}) = x_2^3 x_3 – K T_2 \left( \frac{500}{u x_1 [\sigma_H]} \right)^2 \le 0
$$
6. Worm shaft rigidity constraint
The maximum deflection of the worm shaft under load should not exceed m/50. The deflection caused by tangential and radial forces is:
$$
y = \frac{F_t^2 + F_r^2}{48 E J} L \le \frac{m}{50}
$$
Using the relationships:
- Span length L = 0.9 d2 = 0.9 m u z1
- Moment of inertia J = π/64 · df4 = π/64 · m4 (q − 2.4)4
- Tangential force Ft = 2T1 / d1 = 2T2/(u m q)
- Radial force Fr = (2T2 tan20°)/(u z1 m)
- Elastic modulus E = 2.1 × 105 MPa
After substituting and simplifying, the rigidity constraint becomes:
$$
g_{10}(\mathbf{X}) = 5498 \, x_2^5 (x_3 – 2.4)^4 – T_2 \sqrt{ \left( \frac{x_1}{x_3} \right)^2 + \tan^2 20^\circ } \le 0
$$
Note: The bending strength of worm wheel teeth is usually not critical in enclosed drives, so no separate bending constraint is included.
Summary of Optimization Model
The overall optimization problem is:
$$
\begin{aligned}
\text{minimize } & f(\mathbf{X}) \\
\text{subject to } & g_i(\mathbf{X}) \le 0, \quad i = 1,2,\dots,10
\end{aligned}
$$
This is a three‑dimensional nonlinear programming problem with eight boundary constraints and two performance constraints (contact and rigidity).
Optimization Method
The constrained problem is solved using the exterior penalty function method. A sequential unconstrained minimization technique (SUMT) is employed, where the original constrained problem is transformed into a series of unconstrained minimizations of the penalty function:
$$
P(\mathbf{X}, r_k) = f(\mathbf{X}) + r_k \sum_{i=1}^{10} \left[ \max(0, g_i(\mathbf{X})) \right]^2
$$
Here rk is the penalty factor, which increases gradually with each iteration k. Starting from an initial feasible point, the algorithm minimizes P using a direct search method (e.g., Hooke‑Jeeves pattern search). As rk → ∞, the solution converges to the true constrained optimum. The computation is implemented in MATLAB, and the iterative convergence criterion is set to 1×10−6.
Numerical Example
A worm gear drive with the following data is considered:
- Output torque: T2 = 546550 N·mm
- Load factor: K = 1.1
- Gear ratio: u = 26.39
- Permissible contact stress: [σH] = 180 MPa (material: ZCuSn10P1)
- Efficiency: η = 0.85
Substituting into the model yields the objective and constraints. The initial guess is taken from a conventional design: z1 = 2, m = 6.3 mm, q = 8. After 45 iterations, the exterior penalty method converges to the following optimum:
| Parameter | Optimum (continuous) | Rounded design | Conventional design |
|---|---|---|---|
| z1 (x1) | 3.0315 | 3 | 2 |
| m / mm (x2) | 3.5648 | 4 | 6.3 |
| q (x3) | 16.0000 | 16 | 7.936 |
| Objective f(X) / mm3 | 7.2238 × 105 | — | — |
| Ring volume V* / mm3 (rounded) | — | 1.0106 × 106 | 1.2100 × 106 |
The rounded optimum uses z1 = 3, m = 4 mm, q = 16. The resulting ring volume is 1.0106 × 106 mm3, which is 16.5% less than the conventional volume of 1.2100 × 106 mm3. The contact and rigidity constraints are all satisfied at the optimum.
Discussion
The optimization result shows that a smaller module, a larger diameter coefficient, and an increased number of worm threads lead to a more compact worm wheel ring. The worm thread number z1 is near the upper bound (3 instead of 2), which improves the load‑sharing and reduces the required ring volume. The diameter coefficient reaches its upper limit (16), which means a larger worm pitch diameter relative to the module – this reduces the worm deflection and allows a smaller ring size. The module drops from 6.3 mm to 4 mm, decreasing the tooth size but increasing the number of wheel teeth to maintain the gear ratio (z2 = 26.39 × 3 ≈ 79). The constraints on worm wheel tooth number (30–80) and worm tooth number (2–4) are still satisfied.
The optimization method yields a significant reduction in bronze usage. In mass production, such savings can translate into lower material costs. Moreover, the worm gear drive becomes lighter and more compact, which is beneficial for applications where space and weight are critical, such as in robotics and automotive steering systems.
It should be noted that the rigidity constraint (worm deflection) becomes active at the optimum. The deflection limit of m/50 is reached, indicating that further reduction in ring volume would require a stiffer worm shaft or a different bearing arrangement. The contact strength constraint is also active, ensuring that the worm wheel tooth flank stress is exactly at the allowable limit. This is typical for an optimal design – both performance constraints are binding.
Conclusion
In this work, a comprehensive optimization model for worm gear transmission is presented, with the objective of minimizing the volume of the bronze worm wheel ring. Three design variables (worm thread number, module, diameter coefficient) and ten constraints (geometric, contact strength, and worm shaft rigidity) are considered. Using the exterior penalty function method, an optimal solution is obtained for a practical example. The optimized design reduces the ring volume by 16.5% compared to the conventional design, while satisfying all strength and rigidity requirements. The approach provides a systematic and efficient tool for worm gear designers to achieve cost‑effective and compact transmissions.
Future work can include multi‑objective optimization (e.g., weight together with efficiency), consideration of thermal effects, and integration of manufacturing constraints (e.g., hobbing cutter limitations). The methodology can also be extended to right‑angle worm gear drives and to worm gear units with multiple reductions.