As an engineer specializing in gear design and manufacturing, I have extensively studied the planar internal gear primary-enveloping crown worm drive, a novel type of worm drive that offers high power density and compact structure. This drive consists of an internal gear and a crown worm generated by enveloping the internal gear surface. The selection of design parameters is critical to achieving superior meshing performance, including high load capacity and efficient lubrication. In this article, I will present a comprehensive optimization approach based on macroscopic and microscopic meshing performance, utilizing genetic algorithms to determine optimal parameters. The results demonstrate significant improvements in meshing characteristics, and prototype testing confirms the drive’s high load capacity and efficiency. This work is particularly relevant for internal gear manufacturers seeking to develop advanced transmission systems for applications in aerospace, marine, and other heavy-duty fields where space and weight are constrained.
The planar internal gear primary-enveloping crown worm drive inherits the advantages of enveloping hourglass worm drives, such as multi-tooth contact, high overlap ratio, and long contact lines, while its internal meshing configuration provides a more compact design. For internal gear manufacturers, optimizing this drive involves balancing parameters like the母plane inclination angle, base circle radius, and module to enhance both macro and micro meshing performance. I will begin by establishing the mathematical model based on differential geometry and spatial meshing theory, then proceed to the optimization methodology, and finally validate the results through experimental testing.
Mathematical Model
To analyze the meshing performance of the planar internal gear primary-enveloping crown worm drive, I developed a coordinate system and derived the meshing equations. The process involves enveloping the crown worm surface using the internal gear as the tool gear. The coordinate systems are set up as follows: fixed frames for the worm and internal gear, along with moving frames attached to each component. The key parameters include the母plane inclination angle $\beta$, base circle radius $r_b$, center distance $a$, shaft angle $\delta$, and module $m_t$. The meshing function $\Phi$ is derived from the relative motion between the worm and internal gear.
The meshing function is given by:
$$ \Phi = M_1 \sin \phi_2 + M_2 \cos \phi_2 + M_3 $$
where:
$$ M_1 = v \cos \delta – r_b \sin \delta \cos \beta $$
$$ M_2 = u \cos \delta \cos \beta + a \sin \delta \cos \beta – u \sin \delta \sin \beta $$
$$ M_3 = -a \cos \delta \cos \beta \cos \beta – u i_{21} \cos \beta \sin \beta $$
Here, $u$ and $v$ are parameters defining the母plane, $\phi_2$ is the rotation angle of the internal gear, and $i_{21}$ is the transmission ratio. The contact lines on the internal gear surface are described by:
$$ \mathbf{r}_2 = x_2 \mathbf{i}_2 + y_2 \mathbf{j}_2 + z_2 \mathbf{k}_2 $$
$$ x_2 = r_b \sin \phi_2 – v \cos \phi_2 $$
$$ y_2 = -u $$
$$ z_2 = -v \sin \beta $$
$$ \Phi = 0 $$
The second-kind boundary function $\Phi_t$ and first-kind boundary function $\Psi$ are used to analyze the meshing limits and induced normal curvature. The relative entrainment velocity $v_n$ and lubrication angle $\theta_v$ are critical for assessing the micro meshing performance. The induced normal curvature $k_\delta$ along the contact line normal direction is calculated as:
$$ k_\delta = \frac{(\omega_{py}^{(12)})^2 + (\omega_{px}^{(12)})^2}{\Psi} $$
where $\omega_{px}^{(12)}$ and $\omega_{py}^{(12)}$ are the relative angular velocity components. The lubrication angle $\theta_v$ is given by:
$$ \theta_v = \arcsin \left( \frac{v_{px}^{(12)} \omega_{py}^{(12)} – v_{py}^{(12)} \omega_{px}^{(12)} }{\sqrt{(v_{px}^{(12)})^2 + (v_{py}^{(12)})^2} \sqrt{(\omega_{px}^{(12)})^2 + (\omega_{py}^{(12)})^2}} \right) $$
And the relative entrainment velocity $v_n$ is:
$$ v_n = \frac{v_{px}^{\Sigma} \omega_{py}^{(12)} – v_{py}^{\Sigma} \omega_{px}^{(12)} }{2 \sqrt{(\omega_{px}^{(12)})^2 + (\omega_{py}^{(12)})^2}} $$
These equations form the foundation for evaluating the meshing performance and optimizing the design parameters.
Parameter Optimization Design
In optimizing the planar internal gear primary-enveloping crown worm drive, I focus on selecting design variables that maximize load capacity and lubrication performance while satisfying constraints such as avoiding undercutting and ensuring strength. The optimization involves both macroscopic and microscopic meshing performance criteria.
Design Variables
The primary design variables are the母plane inclination angle $\beta$, base circle radius $r_b$, and module $m_t$. Thus, the vector of design variables is:
$$ \mathbf{x} = [\beta, r_b, m_t]^T $$
These parameters directly influence the contact pattern, induced curvature, and lubrication conditions, making them crucial for internal gear manufacturers to control.
Objective Functions
The objective functions are formulated to enhance both macro and micro meshing performance. For macroscopic performance, I aim to maximize the contact area, uniform distribution of contact lines, and number of simultaneously engaged teeth. For microscopic performance, I minimize the induced normal curvature and maximize the relative entrainment velocity and lubrication angle.
Macroscopic Performance Objectives:
- Contact area $S$: Maximized to distribute load.
- Uniformity of contact lines: Minimize the variance of distances between adjacent contact lines at the tip and root.
- Number of simultaneously engaged teeth $i$: Maximized to increase load sharing.
The objective functions for macroscopic performance are:
$$ f_1(\mathbf{x}) = \frac{1}{S} $$
$$ f_2(\mathbf{x}) = D(m) + D(n) $$
$$ f_3(\mathbf{x}) = \frac{1}{i} $$
where $D(m)$ and $D(n)$ are the variances of distances at the tip and root, respectively.
Microscopic Performance Objectives:
- Induced normal curvature $k_\delta$: Minimized to reduce contact stress.
- Relative entrainment velocity $v_n$: Maximized to improve elastohydrodynamic lubrication.
- Lubrication angle $\theta_v$: Kept close to 90° for optimal lubricant entrainment.
Based on elastohydrodynamic lubrication theory, the minimum film thickness $h_0$ is proportional to $(v_n^{0.7} / k_\delta^{0.43})$. Thus, the objective functions are:
$$ f_4(\mathbf{x}) = \frac{k_\delta^{0.43}}{v_n^{0.7}} $$
$$ f_5(\mathbf{x}) = \left| \frac{\pi}{2} – \theta_v \right| $$
Unified Objective Function:
I combine these objectives using a linear weighted sum method to form a unified objective function:
$$ \min F(\mathbf{x}) = \sum_{j=1}^{5} \alpha_j \frac{f_j(\mathbf{x})}{\chi_j} $$
where $\alpha_j$ are weighting coefficients summing to 1, and $\chi_j$ are scaling factors to normalize the objectives. The weights are chosen based on the importance of each objective, with higher priority given to micro meshing performance.
Constraint Conditions
The optimization must satisfy several constraints to ensure practical feasibility:
- No Secondary Contact: The contact lines should not cross. This is enforced by ensuring the coordinates of adjacent contact lines at the tip and root satisfy:
- No Undercutting on Worm: The root line should not enter the effective worm surface. The constraint is:
- Worm Strength: The root diameter must meet strength requirements, similar to cylindrical worm standards:
- Internal Gear Tip Thickness: The minimum normal tip thickness should exceed 0.25 times the module to ensure strength:
$$ h_1(\mathbf{x}) = \begin{cases}
0 & \text{if } z_{2}^{(m_j)} < z_{2}^{(m_{j+1})} \text{ and } z_{2}^{(n_j)} < z_{2}^{(n_{j+1})} \text{ for } j=1,2,\ldots,i-1 \\
1 & \text{otherwise}
\end{cases} $$
$$ g_1(\mathbf{x}) = r_{f1}^2 – \left( \frac{l}{2} – a \right)^2 – x_{P_c}^2 – y_{P_c}^2 > 0 $$
$$ g_2(\mathbf{x}) = r_{f1}^2 – \left( \frac{l}{2} – a \right)^2 – (m_t q_{\min})^2 > 0 $$
$$ g_3(\mathbf{x}) = s_{a,\min} – 0.25 m_t > 0 $$
These constraints are handled using penalty functions in the optimization algorithm.
Optimization Design Example
To demonstrate the optimization process, I applied it to a drive with fixed center distance and transmission ratio. The basic parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Center distance $a$ (mm) | 100 |
| Worm threads $Z_1$ | 1 |
| Internal gear teeth $Z_2$ | 63 |
| Shaft angle $\delta$ (°) | 25 |
| Worm effective length $l$ (mm) | 70 |
| Internal gear width $b$ (mm) | 50 |
| Addendum coefficient $h_a^*$ | 1 |
| Dedendum coefficient $c^*$ | 0.3 |
The initial design parameters, based on traditional methods, were $\beta = 30^\circ$, $r_b = 43$ mm, and $m_t = 4$ mm. The variable bounds were set as $\mathbf{x}_{\min} = [25, 40, 3.5]^T$ and $\mathbf{x}_{\max} = [35, 60, 4.5]^T$.
I used a genetic algorithm with tournament selection, two-point crossover (probability 0.8), Gaussian mutation (probability 0.01), initial population of 50, and maximum generations of 100. The weighting coefficients were $\alpha = [0.15, 0.15, 0.2, 0.25, 0.25]^T$, and scaling factors $\chi = [160, 0.2, 6, 25, 5]^T$. Penalty coefficients were set to 1000 for constraint violations.
After 56 generations, the optimized parameters were $\mathbf{x} = [28.067, 44.724, 4.239]^T$, which were rounded to $\beta = 28^\circ$, $r_b = 45$ mm, and $m_t = 4.25$ mm. A comparison of meshing performance before and after optimization is shown in Table 2.
| Performance Metric | Before Optimization | After Optimization |
|---|---|---|
| Simultaneous contact teeth $i$ | 6 | 7 |
| Contact area $S$ (mm²) | Reference value | Increased |
| Induced normal curvature $k_\delta$ (avg) | Higher | Reduced |
| Relative entrainment velocity $v_n$ (avg) | Lower | Increased |
| Lubrication angle $\theta_v$ (avg) | Farther from 90° | Closer to 90° |
The optimization resulted in improved macro and micro meshing performance, enhancing load capacity and lubrication. This is beneficial for internal gear manufacturers aiming to produce high-performance drives.
Prototype Testing
Based on the optimized parameters, I manufactured a prototype of the planar internal gear primary-enveloping crown worm drive. The assembly included the internal gear mounted on a hub via interference fit and secured with screws, while the worm shaft was supported by tapered roller bearings. The prototype was compact, with a weight reduction of 7.4% and volume reduction of 12.1% compared to a similar planar double-enveloping worm drive.
The prototype was tested on a 55 kW transmission performance test bench, which consisted of a mechanical-electrical closed-loop system with AC motors, sensors, and control units. The testing followed standard procedures, including no-load tests, run-in tests, and load performance tests. After run-in, the contact pattern on the internal gear surface matched the theoretical analysis, showing uniform distribution across multiple teeth.
Under load conditions (input speed 1500 rpm, output torque 1000 N·m), the maximum temperature rise was 89.8°C, and the transmission efficiency ranged from 65% to 70%. The temperature and efficiency curves indicated slightly better performance for one side of the internal gear due to manufacturing tolerances, but overall, the drive demonstrated high load capacity and efficiency. These results validate the optimization approach and highlight the potential for internal gear manufacturers to adopt this drive in demanding applications.
Conclusion
In this study, I developed a comprehensive optimization method for the planar internal gear primary-enveloping crown worm drive, considering both macroscopic and microscopic meshing performance. The mathematical model based on differential geometry and meshing theory allowed for accurate analysis of contact lines, induced curvature, and lubrication parameters. Using genetic algorithms, I optimized the design parameters to achieve a balance between load capacity and efficiency, while satisfying constraints on strength and undercutting.
The optimized parameters resulted in significant improvements: increased number of simultaneously engaged teeth, larger contact area, reduced induced curvature, and enhanced lubrication conditions. Prototype testing confirmed the drive’s high performance, with substantial load capacity and compact design. This work provides a valuable framework for internal gear manufacturers to design and produce advanced worm drives for heavy-duty applications in aerospace, marine, and other industries where space and weight are critical. Future work could focus on refining the manufacturing processes and exploring material optimizations to further enhance performance.