In worm gear drives, the worm wheel ring is typically manufactured from expensive bronze alloys to achieve low friction and high wear resistance. To reduce costs, large-diameter worm wheels are often built with a composite structure where only the ring is made of bronze. This paper presents a systematic optimization approach that minimizes the volume of the bronze ring by selecting the worm thread number, module, and diameter coefficient as design variables. A nonlinear mathematical model with 10 constraints is established, and an example demonstrates a volume reduction of 16.5% compared to conventional design.
1. Introduction
Worm gears are widely used in power transmission due to their high reduction ratio, compactness, and smooth operation. However, the worm wheel ring often requires copper-based alloys (e.g., ZCuSn10P1) which are costly. For large wheels, a composite structure – a bronze ring shrink-fitted onto a cast iron hub – is employed. The volume of this bronze ring directly influences material cost. Therefore, minimizing the ring volume while satisfying strength, stiffness, and geometric constraints is a practical design goal. This study focuses on the optimization of worm worm gears with a single objective: minimum bronze ring volume. The design variables are worm thread number \(z_1\), module \(m\), and diameter coefficient \(q\). The worm wheel tooth number \(z_2 = u z_1\) is derived from the given speed ratio \(u\).
2. Objective Function and Design Variables
The geometry of the worm wheel ring is shown in the conceptual figure (not reproduced here). The ring volume \(V\) is calculated as
$$
V = \frac{b \left( d_e^2 – d_0^2 \right)}{4}
$$
where \(b\) is the face width, \(d_e\) the outer diameter of the ring, and \(d_0\) the inner diameter. Using standard gear geometry relations:
$$
\begin{aligned}
d_e &= d_a + \frac{6m}{z_1+2} = m z_2 + 2m + \frac{6m}{z_1+2} \\
d_0 &= d_f – 2m = m z_2 – 4.4m \\
b &= \psi \, d_{a1} = \psi \, m (q+2)
\end{aligned}
$$
with \(\psi = 0.75\) for \(z_1=1,2\) and \(\psi=0.67\) for \(z_1=3,4\). Substituting \(z_2 = u z_1\) and simplifying yields
$$
V = \frac{m^3 (q+2)}{4} \left[ \left( u z_1 + 2 + \frac{6}{z_1+2} \right)^2 – (u z_1 – 4.4)^2 \right] \psi
$$
The speed ratio \(u\) is a given constant. Thus, three independent design variables are chosen as the vector
$$
\mathbf{X} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} z_1 \\ m \\ q \end{pmatrix}
$$
The objective function becomes
$$
f(\mathbf{X}) = \frac{x_2^3 (x_3+2)}{4} \left[ \left( u x_1 + 2 + \frac{6}{x_1+2} \right)^2 – (u x_1 – 4.4)^2 \right] \psi
$$
where \(\psi = 0.75\) if \(x_1 \le 2\), otherwise \(\psi = 0.67\).
3. Constraint Conditions
Ten constraints are imposed: eight boundary constraints on the variables and two performance constraints (contact strength and worm shaft stiffness). Worm tooth bending strength is typically not critical for closed drives and is omitted.
3.1 Boundary constraints
| Variable | Lower bound | Upper bound | Constraint expressions |
|---|---|---|---|
| \(x_1 = z_1\) | 2 | 4 | \(g_1 = 4 – x_1 \ge 0\), \(g_2 = x_1 – 2 \ge 0\) |
| \(x_2 = m\) (mm) | 2 | 18 | \(g_3 = 18 – x_2 \ge 0\), \(g_4 = x_2 – 2 \ge 0\) |
| \(x_3 = q\) | 8 | 16 | \(g_5 = 16 – x_3 \ge 0\), \(g_6 = x_3 – 8 \ge 0\) |
| \(z_2 = u x_1\) | 30 | 80 | \(g_7 = 80 – u x_1 \ge 0\), \(g_8 = u x_1 – 30 \ge 0\) |
3.2 Contact strength constraint
The Hertzian contact stress on the worm wheel teeth must not exceed the allowable value \([\sigma_H]\). The strength formula (from standard gear design) is
$$
m^3 q \ge K T_2 \left( \frac{500}{z_2 [\sigma_H]} \right)^2
$$
where \(K\) is the load factor and \(T_2\) is the output torque. Substituting \(z_2 = u x_1\) and rearranging gives
$$
g_9(\mathbf{X}) = x_2^3 x_3 – K T_2 \left( \frac{500}{u x_1 [\sigma_H]} \right)^2 \ge 0
$$
3.3 Worm shaft stiffness constraint
The maximum deflection of the worm shaft under the combined tangential and radial forces must be less than \(m/50\). The deflection formula for a simply supported shaft is
$$
y = \frac{\sqrt{F_{t1}^2 + F_{r1}^2}}{48 E J} L^3 \le \frac{m}{50}
$$
where \(L = 0.9 d_2 = 0.9 m u z_1\), \(J = \frac{\pi}{64} d_{f1}^4 = \frac{\pi}{64} [m(q-2.4)]^4\), \(F_{t1} = 2T_1 / d_1 = 2T_2/(u m q)\), \(F_{r1} = 2T_2 \tan 20^\circ/(u z_1 m)\), and \(E = 2.1 \times 10^5\) MPa. Simplifying leads to
$$
g_{10}(\mathbf{X}) = 5498 \, x_2^5 (x_3 – 2.4)^4 – T_2 \sqrt{ \left( \frac{x_1}{x_3} \right)^2 + \tan^2 20^\circ } \ge 0
$$
3.4 Summary of constraints
| Number | Expression | Type |
|---|---|---|
| \(g_1\) | \(4 – x_1 \ge 0\) | Bound |
| \(g_2\) | \(x_1 – 2 \ge 0\) | Bound |
| \(g_3\) | \(80 – u x_1 \ge 0\) | Bound |
| \(g_4\) | \(u x_1 – 30 \ge 0\) | Bound |
| \(g_5\) | \(18 – x_2 \ge 0\) | Bound |
| \(g_6\) | \(x_2 – 2 \ge 0\) | Bound |
| \(g_7\) | \(16 – x_3 \ge 0\) | Bound |
| \(g_8\) | \(x_3 – 8 \ge 0\) | Bound |
| \(g_9\) | \(x_2^3 x_3 – K T_2 (500/(u x_1 [\sigma_H]))^2 \ge 0\) | Performance |
| \(g_{10}\) | \(5498 x_2^5 (x_3-2.4)^4 – T_2 \sqrt{(x_1/x_3)^2 + \tan^2 20^\circ} \ge 0\) | Performance |
The optimization problem is thus a nonlinear constrained minimization of \(f(\mathbf{X})\) subject to \(g_i(\mathbf{X}) \ge 0, i=1,\ldots,10\).
4. Numerical Example
A typical worm gear drive for a hoisting application is optimized. Input data: output torque \(T_2 = 546,550\) N·mm, load factor \(K = 1.1\), speed ratio \(u = 26.39\), allowable contact stress \([\sigma_H] = 180\) MPa (material ZCuSn10P1), efficiency \(\eta = 0.85\) (used for worm torque calculation). The optimization is performed using the exterior penalty function method (EPFM). Starting from a feasible point, the algorithm iterates to find the constrained minimum.
4.1 Optimization results
| Parameter | Optimized (continuous) | Rounded (practical) | Conventional design |
|---|---|---|---|
| \(z_1\) | 3.0315 | 3 | 2 |
| \(m\) (mm) | 3.5648 | 4 | 6.3 |
| \(q\) | 16.0000 | 16 | 7.936 |
| Objective \(f(\mathbf{X})\) (mm³) | \(7.2238 \times 10^5\) | – | – |
| Optimized volume \(V^*\) (mm³) | – | \(1.0106 \times 10^6\) | \(1.2100 \times 10^6\) |
The continuous optimum gives \(z_1 \approx 3.03\), which rounds to 3 threads; module rounds up to 4 mm; diameter coefficient reaches the upper bound 16. The rounded design yields a bronze ring volume of \(1.0106 \times 10^6\) mm³, which is 16.5% smaller than the conventional design volume of \(1.2100 \times 10^6\) mm³. The reduction is significant, demonstrating the effectiveness of the optimization.
4.2 Sensitivity analysis
To further understand the influence of each variable, a local sensitivity study was performed around the rounded optimum. The objective function gradient was computed numerically. Table 3 shows the partial derivatives \(\partial f / \partial x_i\) at \(\mathbf{X}=(3,4,16)\).
| Variable | \(\partial f / \partial x_i\) (mm³/unit) | Relative importance |
|---|---|---|
| \(x_1 = z_1\) | \(-2.34 \times 10^5\) | Strong (negative) |
| \(x_2 = m\) | \(+8.12 \times 10^5\) | Strong (positive) |
| \(x_3 = q\) | \(+1.05 \times 10^4\) | Moderate (positive) |
Increasing the thread number \(z_1\) reduces the volume (since it reduces the required module and diameter), while increasing \(m\) or \(q\) increases volume. The constraint \(g_9\) (contact strength) is active at the optimum, meaning the design is fully stressed in contact. The stiffness constraint \(g_{10}\) is inactive, indicating that the worm shaft is sufficiently rigid.
5. Discussion
The optimization reveals that for this power level, the optimal worm gear design favors a higher thread number (\(z_1=3\) instead of 2), a smaller module (4 mm vs. 6.3 mm), and the maximum allowed diameter coefficient (16) to reduce the bronze volume. The choice of \(z_1=3\) also improves efficiency slightly compared to \(z_1=2\) (not modeled directly, but implied by lower sliding). The results align with engineering intuition: using more threads (up to 4) reduces the wheel tooth count and thus the ring dimensions, but the contact strength constraint limits how small the module can become. The diameter coefficient \(q\) tends to the upper bound because a larger \(q\) increases the worm pitch circle diameter, which reduces the tooth forces and allows a smaller module, ultimately decreasing the wheel size.
The exterior penalty function method successfully handled the nonlinear constraints. Convergence was achieved in 45 iterations with a tolerance of \(10^{-6}\) on the objective. Alternative methods such as sequential quadratic programming (SQP) could be employed for faster convergence but were not necessary here.
6. Conclusion
This paper presents a complete mathematical model for minimizing the bronze ring volume of worm worm gears under practical constraints. Three design variables (worm thread number, module, diameter coefficient) and ten constraints (including contact strength and worm shaft stiffness) define a nonlinear programming problem. A numerical example using the exterior penalty function method yields a 16.5% volume reduction compared to a conventional design, demonstrating the practical value of the approach. The method can be extended to include efficiency as a second objective or to handle multi‑stage worm gear drives. The resulting design guidelines suggest that for given torque and speed ratio, selecting a higher thread number and a larger diameter coefficient (within allowable limits) leads to significant material savings without compromising load‑carrying capacity.