In this study, I present a multi-objective fuzzy optimization design approach for cylindrical worm gear drives. The primary goal is to reduce the manufacturing cost and improve the transmission efficiency, which are common challenges in traditional worm gear design. By employing fuzzy optimization theory, I establish two objective functions: minimizing the volume of the worm wheel rim and maximizing the transmission efficiency (i.e., minimizing the reciprocal of efficiency). The design variables are the number of worm gear starts, module, and diameter coefficient. Constraints include geometric boundaries and performance requirements, such as contact stress, bending stress, shaft deflection, sliding velocity, and thermal balance. The fuzzy constraints are processed using membership functions and a non-fuzzy transformation based on an optimal level cut. The optimization is carried out using Matlab programming with its Optimization Toolbox. The results show that the optimized worm gear drive achieves a 29.3% reduction in worm wheel rim volume and a 2.34% increase in efficiency compared to a conventional design. This work demonstrates the effectiveness of integrating fuzzy logic with Matlab for practical worm gear design.
1. Introduction
Worm gear drives are widely used in mining, metallurgy, transportation, and lifting machinery due to their large transmission ratio, compact structure, smooth operation, and low noise. However, conventional worm gear design often relies on empirical rules and iterative trial calculations, which are time-consuming and rarely yield optimal results. To address these limitations, I adopt a multi-objective fuzzy optimization methodology. The cylindrical worm gear is selected as the research object because of its typical use in power transmission. By combining fuzzy set theory with mathematical programming, I can handle the inherent ambiguity in engineering constraints and achieve a balanced trade-off between conflicting objectives. The use of Matlab’s optimization toolbox simplifies the computational process and makes the design procedure accessible for engineering practice.
2. Fuzzy Optimization Mathematical Model
2.1 Design Variables
I choose three independent parameters that significantly influence the performance and cost of the worm gear drive: the number of worm gear starts \(z_1\), the module \(m\), and the diameter coefficient \(q\). These variables form the design vector:
\[
\mathbf{X} = [x_1, x_2, x_3]^T = [z_1, m, q]^T
\]
2.2 Objective Functions
2.2.1 Minimizing Worm Wheel Rim Volume
The worm wheel rim is typically made of expensive tin bronze; therefore, minimizing its volume reduces material cost. The rim volume \(f_1(\mathbf{X})\) is calculated based on geometric dimensions:
\[
f_1(\mathbf{X}) = \frac{\pi}{4} (d_e^2 – d_0^2) B
\]
where:
\[
d_e = d_{a2} + m = m (i z_1 + 3)
\]
\[
d_0 = d_{f2} – 2e = m (i z_1 – 6.4)
\]
\[
B = 0.67 d_{a1} = 0.67 m (q + 2)
\]
Substituting and simplifying yields:
\[
f_1(\mathbf{X}) = 0.52622 x_2^3 (x_3 + 2) [(20 x_1 + 3)^2 – (20 x_1 – 6.4)^2]
\]
2.2.2 Maximizing Transmission Efficiency
The transmission efficiency \(\eta\) of a worm gear is typically low (0.75–0.92). I aim to maximize \(\eta\), which is equivalent to minimizing its reciprocal \(f_2(\mathbf{X})\). The efficiency is given by:
\[
\eta = \frac{0.96 \tan \lambda}{\tan(\lambda + \rho_v)}
\]
so:
\[
f_2(\mathbf{X}) = \frac{1}{\eta} = \frac{\tan(\lambda + \rho_v)}{0.96 \tan \lambda}
\]
Here, \(\lambda = \arctan(z_1/q)\) is the lead angle, and \(\rho_v\) is the equivalent friction angle. The equivalent friction coefficient \(f_v\) is obtained by curve fitting from standard tables using Matlab:
\[
f_v = 0.0175192 \, e^{1.0460682 / v_s}
\]
with sliding velocity:
\[
v_s = \frac{m n_1}{19100} \sqrt{z_1^2 + q^2}
\]
where \(n_1\) is the worm gear rotational speed (given as 960 rpm in the case study). After substitution and simplification:
\[
f_2(\mathbf{X}) = \frac{x_1 x_3 + f_v x_3^2}{0.96 x_1 x_3 – 0.96 x_1^2 f_v}
\]
2.2.3 Unified Objective Function
Since the two objectives have different magnitudes, I use a weighted sum approach. The first objective \(f_1\) is on the order of \(10^6\) mm³, while \(f_2\) is dimensionless. To balance them, I multiply \(f_2\) by \(10^6\) and assign weights \(\omega_1 = 0.8\) and \(\omega_2 = 0.2\). The unified objective function becomes:
\[
f(\mathbf{X}) = 0.8 f_1(\mathbf{X}) + 0.2 \times 10^6 f_2(\mathbf{X})
\]
\[
= 0.420976 x_2^3 (x_3 + 2) [(20 x_1 + 3)^2 – (20 x_1 – 6.4)^2] – \frac{4.8 \times 10^6 (x_1 x_3 + f_v x_3^2)}{x_1 (x_3 – x_1 f_v)}
\]
2.3 Constraint Conditions
Both geometric and performance constraints exhibit fuzziness, meaning there is a gradual transition from fully acceptable to fully unacceptable. The constraints for this worm gear drive (transmission ratio \(i = 20\), power 7.5 kW) are listed below.
| Type | Constraint Expression | Lower/Upper Bounds |
|---|---|---|
| Boundary | Number of starts: \(2 \le g_1 = x_1 \le 6\) | 2–6 |
| Number of worm wheel teeth: \(30 \le g_2 = 20 x_1 \le 80\) | 30–80 | |
| Module: \(2 \le g_3 = x_2 \le 18\) (mm) | 2–18 | |
| Diameter coefficient: \(8 \le g_4 = x_3 \le 15\) | 8–15 | |
| Performance | Contact stress: \(\sigma_H \le [\sigma_H]\) | \([\sigma_H] = 166.57\) MPa |
| Bending stress: \(\sigma_F \le [\sigma_F]\) | \([\sigma_F] = 38.164\) MPa | |
| Shaft deflection: \(f \le [f] = m/50\) | – | |
| Sliding velocity: \(v_s \le [v_s] = 16\) m/s | – | |
| Heat balance: \(\Delta t \le [\Delta t] = 60^\circ\)C | – |
The detailed formulas for the performance constraints are:
\[
g_5(\mathbf{X}) = \frac{28066.176}{\sqrt{x_1^2 x_2^3 x_3}} \le [\sigma_H]
\]
\[
g_6(\mathbf{X}) = \frac{119866.869 \, e^{-0.049 x_1} \left(1 – \frac{\arctan(x_1/x_3)}{120}\right)}{x_1 x_2^3 x_3} \le [\sigma_F]
\]
\[
g_7(\mathbf{X}) = \frac{3.411852 \times 10^{-4} (20 x_1 + 3)^3 \sqrt{\left(\frac{0.36397}{x_1}\right)^2 + \left(\frac{1.17647}{x_3}\right)^2}}{(x_3 – 2.4)^4 x_2^2} \le [f] = \frac{x_2}{50}
\]
\[
g_8(\mathbf{X}) = 0.05026178 \, x_2 \sqrt{x_1^2 + x_3^2} \le [v_s]
\]
\[
g_9(\mathbf{X}) = \frac{2417407.226}{[x_2 (x_3 + 20 x_1)]^{1.75}} \le [\Delta t]
\]
3. Fuzzy Membership Functions and Non-Fuzzy Transformation
3.1 Membership Functions
I adopt linear membership functions for both boundary and performance constraints. For a performance constraint \(g_j(\mathbf{X}) \le \sigma_j\), the membership function is:
\[
\mu_{g_j}(\mathbf{X}) =
\begin{cases}
1 & g_j(\mathbf{X}) \le \sigma_j^l \\[4pt]
\frac{\sigma_j^u – g_j(\mathbf{X})}{\sigma_j^u – \sigma_j^l} & \sigma_j^l \le g_j(\mathbf{X}) \le \sigma_j^u \\[4pt]
0 & g_j(\mathbf{X}) \ge \sigma_j^u
\end{cases}
\]
where \(\sigma_j^l\) and \(\sigma_j^u\) are the lower and upper limits of the fuzzy allowable interval. For a boundary constraint \(x_j\) with allowed range \([x_j^l, x_j^u]\), the membership function is triangular:
\[
\mu_{x_j} =
\begin{cases}
\frac{x_j – x_j^{ll}}{x_j^l – x_j^{ll}} & x_j^{ll} \le x_j \le x_j^l \\[4pt]
1 & x_j^l \le x_j \le x_j^u \\[4pt]
\frac{x_j^{uu} – x_j}{x_j^{uu} – x_j^u} & x_j^u \le x_j \le x_j^{uu} \\[4pt]
0 & \text{otherwise}
\end{cases}
\]
3.2 Determination of Fuzzy Intervals
I use the expansion coefficient method. For performance constraints, the upper bound expansion coefficient \(\beta^-\) is taken as 1.05, and for the lower bound \(\beta^-\) as 0.85. The resulting fuzzy intervals are listed in Table 2.
| Constraint | Lower Limit (ll) | Lower Bound (l) | Upper Bound (u) | Upper Limit (uu) |
|---|---|---|---|---|
| \([\sigma_H]\) (MPa) | – | 166.6 | 174.9 | – |
| \([\sigma_F]\) (MPa) | – | 38.16 | 40.07 | – |
| \([f]\) (mm) | – | 0.02 \(x_2\) | 0.021 \(x_2\) | – |
| \([v_s]\) (m/s) | – | 16 | 16.8 | – |
| \([\Delta t]\) (°C) | – | 60 | 63 | – |
| \(z_1\) | 1.7 | 2 | 6 | 6.3 |
| \(m\) (mm) | 1.7 | 2 | 18 | 18.9 |
| \(q\) | 6.8 | 8 | 15 | 15.75 |
| \(z_2\) | 25.5 | 30 | 80 | 84 |
3.3 Optimal Level Cut \(\lambda^*\)
I determine the optimal level cut using a fuzzy comprehensive evaluation method. After considering factor sets, evaluation indices, and weight sets, I obtain \(\lambda^* = 0.6\). This value is used to defuzzify the constraints into crisp inequalities.
3.4 Non-Fuzzy Optimization Model
With \(\lambda^* = 0.6\), the fuzzy constraints are converted to deterministic ones. The resulting non-fuzzy model is:
\[
\min f(\mathbf{X}) \quad \text{subject to}
\]
\[
1.88 \le x_1 \le 6.12
\]
\[
28.2 \le 20 x_1 \le 81.6
\]
\[
1.88 \le x_2 \le 18.36
\]
\[
7.52 \le x_3 \le 15.35
\]
\[
g_5(\mathbf{X}) \le 169.92
\]
\[
g_6(\mathbf{X}) \le 38.924
\]
\[
g_7(\mathbf{X}) \le 0.0204 x_2
\]
\[
g_8(\mathbf{X}) \le 16.32
\]
\[
g_9(\mathbf{X}) \le 61.2
\]
4. Matlab Implementation and Optimization Results
4.1 Objective Function Function File
I create a Matlab function myobj.m that computes the unified objective:
function f = myobj(x)
vs = 0.05026178 * x(2) * sqrt(x(1)^2 + x(3)^2);
fv = 0.0175192 * exp(1.0460682 / vs);
term1 = 0.420976 * x(2)^3 * (x(3) + 2) * ((20*x(1)+3)^2 - (20*x(1)-6.4)^2);
term2 = (4.8e6 * (x(1)*x(3) + fv*x(3)^2)) / (x(1)*(x(3) - x(1)*fv));
f = term1 - term2;
end4.2 Nonlinear Constraint Function File
I define mycon.m for the five performance constraints:
function [c, ceq] = mycon(x)
c(1) = 28066.176 / sqrt(x(1)^2 * x(2)^3 * x(3)) - 169.92;
c(2) = 119866.869 * exp(-0.049*x(1)) * (1 - atan(x(1)/x(3))/120) / (x(1)*x(2)^3*x(3)) - 38.924;
c(3) = 3.411852e-4 * (20*x(1)+3)^3 * sqrt((0.36397/x(1))^2 + (1.17647/x(3))^2) / ((x(3)-2.4)^4 * x(2)^2) - 0.0204*x(2);
c(4) = 0.05026178 * x(2) * sqrt(x(1)^2 + x(3)^2) - 16.32;
c(5) = 2417407.226 / (x(2)*(x(3)+20*x(1)))^1.75 - 61.2;
ceq = [];
end4.3 Main Optimization Script
I use the fmincon solver with the interior-point algorithm:
lb = [1.7, 1.7, 6.8];
ub = [6.3, 18.9, 15.75];
x0 = [1.7, 1.7, 6.8];
A = [-1 0 0; 1 0 0; 0 -1 0; 0 1 0; 0 0 -1; 0 0 1];
b = [-1.88, 4.08, -1.88, 18.36, -7.52, 15.35];
options = optimset('Display','iter','Algorithm','interior-point');
[x_opt, fval] = fmincon(@myobj, x0, A, b, [], [], lb, ub, @mycon, options);4.4 Optimization Results
Running the script yields the continuous optimum: \(x_1 = 4.08\), \(x_2 = 4.743\) mm, \(x_3 = 15.35\). These values are rounded to standard values: \(z_1 = 4\), \(m = 5\) mm, \(q = 15\).
For comparison, the traditional design parameters were \(z_1 = 2\), \(m = 8\) mm, \(q = 10\). Table 3 summarizes the performance comparison.
| Parameter | Traditional | Optimized | Improvement |
|---|---|---|---|
| Number of starts \(z_1\) | 2 | 4 | – |
| Module \(m\) (mm) | 8 | 5 | – |
| Diameter coefficient \(q\) | 10 | 15 | – |
| Worm wheel rim volume (mm³) | 2.327958 × 10⁶ | 1.64606 × 10⁶ | −29.3% |
| Transmission efficiency \(\eta\) | 0.8586 | 0.8787 | +2.34% |
The optimized worm gear drive achieves a significant volume reduction of 29.3% while simultaneously increasing the efficiency by 2.34%. This demonstrates that the multi-objective fuzzy optimization method effectively balances the conflicting goals of cost and performance.
5. Conclusion
In this work, I have presented a comprehensive multi-objective fuzzy optimization procedure for cylindrical worm gear drives using Matlab. By formulating the problem with two objectives—minimizing worm wheel rim volume and maximizing transmission efficiency—and incorporating fuzzy constraints that reflect the inherent uncertainty in engineering design, I obtained a solution that significantly outperforms the traditional trial-and-error approach. The optimized worm gear design not only reduces material cost by nearly 30% but also improves efficiency. The integration of fuzzy set theory with Matlab’s optimization toolbox provides a practical and efficient tool for gear designers. Future work could extend this methodology to other types of worm gears or include additional performance criteria such as noise and vibration.
Keywords: worm gear, cylindrical worm drive, multi-objective optimization, fuzzy optimization, Matlab, transmission efficiency, volume minimization.