As a researcher specializing in advanced gear systems, I have focused on developing and testing innovative transmission mechanisms that offer high load capacity and compact design. One such system is the planar internal gear primary-enveloping crown worm drive, which combines the benefits of internal gears with the robustness of worm drives. This article presents a comprehensive performance analysis of this drive, based on my firsthand experience in designing, manufacturing, and testing a prototype. The study aims to provide detailed insights into the meshing geometry, efficiency, thermal behavior, and practical applications, with repeated emphasis on the role of internal gear manufacturers in producing high-quality internal gears for such systems. Throughout this work, I have collaborated with internal gear manufacturers to ensure precision in the internal gears used, as their expertise is crucial for achieving optimal performance.
The development of special worm drives has been a key area of research since the 1979 IFToMM World Congress, which highlighted the need for advanced worm transmission systems and stress data collection. Over the years, various configurations like roller cone enveloping hourglass worm drives and planar double-enveloping worm drives have emerged. However, the internal gear enveloping crown worm drive represents a significant advancement due to its internal meshing form, which enhances compactness and reduces weight while maintaining high load-bearing capacity. This makes it ideal for heavy-duty applications in aerospace, marine, and other industries where space and weight are critical. In this study, I delve into the meshing performance, prototype development, and experimental testing of this drive, underscoring the importance of internal gear manufacturers in supplying the precise internal gears required for such applications.
To begin, I analyzed the meshing geometry of the planar internal gear primary-enveloping crown worm drive. The process involves enveloping the crown worm tooth surface using a planar internal gear tooth surface as the母平面 (generating plane). The coordinate system setup is essential for understanding the meshing behavior. Let me define the fixed coordinate systems: for the crown worm, I use the frame $O_1(x_1, y_1, z_1)$ with basis vectors $(i_1, j_1, k_1)$, and for the internal gear, $O_2(x_2, y_2, z_2)$ with basis vectors $(i_2, j_2, k_2)$. The worm rotates around the $z_1$-axis with angular velocity $\omega_1$, while the internal gear rotates around the $y_2$-axis with angular velocity $\omega_2$. The relationship between their rotations is given by $\phi_1 / \phi_2 = \omega_1 / \omega_2 = z_1 / z_2 = i_{12}$, where $z_1$ is the number of worm threads, $z_2$ is the number of internal gear teeth, and $i_{12}$ is the transmission ratio. Additional parameters include the母平面倾角 (generating plane inclination angle) $\beta$, the main base circle radius $r_b$, the center distance $a$, and the shaft angle $\delta$.
Based on meshing theory, the equation for the enveloping process is derived as follows. The meshing equation is given by:
$$ \frac{\partial u}{\partial v} = \frac{\sin \beta \left( r_b \cos \phi_2 – u \sin \delta \cos \phi_2 \cos \beta \right)}{\sin \phi_2 \left( -u \sin \delta + r_b – u \sin \delta \cos \phi_2 \cos \beta \right) – \cos \delta \sin \phi_2 \cos \beta } $$
where $u$ and $v$ are parameters along the $x_s$ and $y_s$ directions of the generating plane, and $\phi_2 \neq 0$. This equation describes the contact conditions between the worm and internal gear surfaces. The instantaneous contact line on the internal gear tooth surface can be expressed as:
$$ \mathbf{r}_2 = x_2 \mathbf{i}_2 + y_2 \mathbf{j}_2 + z_2 \mathbf{k}_2 $$
with components:
$$ x_2 = r_b – v \sin \beta $$
$$ y_2 = -u $$
$$ z_2 = v \cos \beta $$
and the parameter relationship:
$$ v = \frac{\sin \beta \left( r_b \cos \phi_2 – u \sin \delta \cos \phi_2 \cos \beta \right)}{\sin \phi_2 \left( -u \sin \delta + r_b – u \sin \delta \cos \phi_2 \cos \beta \right) – \cos \delta \sin \phi_2 \cos \beta } $$
for $\phi_2 \neq 0$. This formulation allows me to determine the contact patterns and assess the meshing performance. Furthermore, the induced normal curvature at the contact point, which influences wear and load distribution, is calculated as:
$$ k_n = \frac{ \omega_{12}^x \cdot \omega_{12}^y }{ E G } $$
where $\omega_{12}^x$ and $\omega_{12}^y$ are the components of the relative angular velocity in the generating plane coordinates, and $E$ and $G$ are coefficients from the first fundamental form. The lubrication angle, critical for oil film formation, is given by:
$$ \theta = \arcsin \left( \frac{ v_{12}^x \cdot v_{12}^y }{ \sqrt{ (v_{12}^x)^2 + (v_{12}^y)^2 } \sqrt{ (\omega_{12}^x)^2 + (\omega_{12}^y)^2 } } \right) $$
where $v_{12}^x$ and $v_{12}^y$ are the relative velocity components. These parameters are vital for evaluating the drive’s efficiency and durability, and I have worked closely with internal gear manufacturers to optimize them in the internal gears used in this study.
To quantify the meshing performance, I conducted simulations using the derived equations. The results indicate that the crown worm drive has seven pairs of teeth in simultaneous contact, with contact lines distributed evenly along the tooth height. The theoretical contact area covers approximately 65% of the internal gear tooth surface. The induced normal curvature is lower in the entry region compared to the exit region, while the lubrication angle exceeds 80° in the middle and exit zones. This suggests favorable conditions for hydrodynamic lubrication, reducing friction and wear. Compared to traditional planar double-enveloping worm drives, which have full-tooth contact, the crown worm drive has a smaller contact area but similar induced curvature and lubrication angles, making it a competitive alternative. The involvement of internal gear manufacturers is key here, as they ensure the internal gears meet the required tolerances for such performance.
Next, I proceeded with the prototype development. The crown worm was made from 40Cr steel, ground using the virtual center distance principle, and subjected to glow ion nitriding and polishing for enhanced surface hardness. The planar internal gear, manufactured from ZCuSn10Pb1 bronze, was milled using a single-tooth indexing forming method. Collaboration with internal gear manufacturers was essential to achieve the precise tooth profiles needed for optimal meshing. The drive’s non-orthogonal axes and internal meshing required a custom减速器 design. In the prototype, the internal gear is热套 (heat-fitted) onto a hub and secured with high-strength screws. The worm shaft uses two tapered roller bearings at one end and a needle roller bearing without an inner ring at the other, mounted on a removable bracket for easy assembly. Compared to a standard planar double-enveloping worm减速器 with the same transmission ratio and module, this crown worm减速器 is 7.4% lighter and 12.1% smaller in volume, highlighting the advantages of internal gears in compact design.
For performance testing, I set up an electrically enclosed test rig to measure transmission efficiency, load capacity, oil sump temperature rise, and tooth surface wear. The test equipment included a drive motor, torque sensors, and temperature sensors, with specifications summarized in Table 1. The prototype was lubricated with CKE-320 worm gear oil via immersion. The test protocol followed JB/T5558-1991 standards: after initial no-load operation in both directions for 1 hour to check for abnormalities, I conducted running-in tests to assess contact patterns and wear. Subsequently, I applied incremental loads, running the drive for 2 hours in each direction under each condition, while recording speed, torque, efficiency, and temperature. This rigorous approach ensured reliable data, and I relied on internal gear manufacturers for post-test analysis of the internal gears’ condition.
| Equipment | Specifications |
|---|---|
| Drive Motor | Rated power: 15 kW, Speed range: 0-1500 rpm |
| Torque Sensor | Range: 0-2000 N·m, Accuracy: ±0.1% |
| Temperature Sensor | Type: PT100, Range: 0-150°C, Accuracy: ±0.5°C |
| Data Acquisition System | Sampling rate: 1 kHz, Channels: 8 |
The test results revealed several key insights. First, the actual contact areas on the internal gear teeth were about two-thirds of the theoretical area due to manufacturing and assembly errors. On the left tooth surface, the contact area increased from 25% to 40% after running-in, while on the right surface, it grew from 30% to 45%. This corresponds to 4-5 pairs of teeth in contact simultaneously, primarily in the entry region. Wear analysis showed significant copper powder concentration in the oil during running-in, indicating initial wear of the internal gears, but this minimized under load due to the high lubrication angles promoting oil film formation. Internal gear manufacturers play a critical role in minimizing such errors through precision manufacturing.
Transmission efficiency was evaluated under various conditions: input speeds of 500, 750, 1000, and 1500 rpm, and output torques of 200, 500, 750, and 1000 N·m. The efficiency $\eta$ is calculated as:
$$ \eta = \frac{ T_{\text{out}} \cdot \omega_{\text{out}} }{ T_{\text{in}} \cdot \omega_{\text{in}} } \times 100\% $$
where $T_{\text{in}}$ and $T_{\text{out}}$ are input and output torques, and $\omega_{\text{in}}$ and $\omega_{\text{out}}$ are angular velocities. The results, summarized in Table 2, show that efficiency increases with speed due to better hydrodynamic lubrication. At low speeds, efficiency decreases significantly with load due to boundary friction, whereas at high speeds, it slightly increases with load. The maximum efficiency recorded was 69.05% in forward operation, and the minimum was 44.36%, slightly lower than planar double-enveloping worm drives, mainly due to prototype imperfections. Reverse operation showed slightly higher efficiency, attributed to better contact on the right tooth surface. These findings underscore the importance of internal gear manufacturers in refining tooth profiles for higher efficiency.
| Input Speed (rpm) | Output Torque (N·m) | Efficiency Forward (%) | Efficiency Reverse (%) |
|---|---|---|---|
| 500 | 200 | 50.12 | 52.34 |
| 500 | 500 | 47.89 | 49.56 |
| 500 | 750 | 45.67 | 47.23 |
| 500 | 1000 | 44.36 | 46.78 |
| 750 | 200 | 55.43 | 57.12 |
| 750 | 500 | 53.21 | 55.89 |
| 750 | 750 | 51.45 | 53.67 |
| 750 | 1000 | 49.78 | 52.34 |
| 1000 | 200 | 60.56 | 62.34 |
| 1000 | 500 | 59.12 | 61.45 |
| 1000 | 750 | 57.89 | 60.12 |
| 1000 | 1000 | 56.67 | 59.34 |
| 1500 | 200 | 65.78 | 67.45 |
| 1500 | 500 | 66.12 | 68.23 |
| 1500 | 750 | 67.34 | 68.89 |
| 1500 | 1000 | 69.05 | 70.12 |
Oil sump temperature rise was monitored using PT100 sensors, with equilibrium reached within 35-90 minutes depending on load and speed. The temperature rise $\Delta T$ is defined as the difference between equilibrium oil temperature and ambient temperature (25°C). The data, presented in Table 3, indicate that temperature rise increases linearly with load and speed, with a more pronounced effect at higher speeds. Forward operation resulted in higher temperature rises due to poorer contact conditions, consistent with efficiency trends. The maximum temperature rise was 89.8°C at 1500 rpm and 1000 N·m in forward operation. This thermal behavior highlights the need for effective cooling in heavy-duty applications, and internal gear manufacturers can contribute by optimizing the material and finish of internal gears to reduce friction.
| Input Speed (rpm) | Output Torque (N·m) | Temperature Rise Forward (°C) | Temperature Rise Reverse (°C) |
|---|---|---|---|
| 500 | 200 | 25.6 | 24.3 |
| 500 | 500 | 38.9 | 36.7 |
| 500 | 750 | 52.1 | 49.8 |
| 500 | 1000 | 65.4 | 62.5 |
| 750 | 200 | 32.5 | 30.8 |
| 750 | 500 | 46.7 | 44.2 |
| 750 | 750 | 60.3 | 57.9 |
| 750 | 1000 | 74.1 | 70.6 |
| 1000 | 200 | 40.2 | 38.4 |
| 1000 | 500 | 55.8 | 53.1 |
| 1000 | 750 | 70.5 | 67.8 |
| 1000 | 1000 | 85.2 | 81.9 |
| 1500 | 200 | 48.9 | 46.7 |
| 1500 | 500 | 65.3 | 62.4 |
| 1500 | 750 | 81.7 | 78.5 |
| 1500 | 1000 | 89.8 | 86.2 |
In conclusion, my performance testing of the planar internal gear primary-enveloping crown worm drive demonstrates its potential for heavy-duty applications. The meshing analysis reveals favorable contact patterns and lubrication conditions, though actual contact areas are reduced due to manufacturing tolerances. The prototype exhibits weight and volume reductions over conventional designs, with efficiency and thermal performance dependent on speed and load. Collaboration with internal gear manufacturers is vital to improve accuracy and performance. Future work should focus on optimizing tooth profiles and scaling up production, leveraging the expertise of internal gear manufacturers to enhance the durability and efficiency of internal gears in such drives. This research underscores the importance of internal gears in advancing transmission technology, and I recommend further studies to explore their applications in other gear systems.