In my research on advanced transmission systems for robotic applications, I focused on the design of a novel beveloid internal gear plane enveloping external-rotor drum-worm transmission device. The primary objective was to develop a compact, high-precision reducer that could be utilized in robotic joints, thereby contributing to the localization of robot reducer technology. By analyzing the internal meshing motion of the worm gears, I established six coordinate frames to mathematically describe the kinematic relationships.
I began by using the tooth surfaces of a beveloid internal gear as the tool surfaces. Through the auxiliary and working frames on the internal gear, I determined the relationships among the worm-wheel rotation angle, the worm rotation angle, and the working angle. Based on the principles of gearing, I established the meshing equations for the drum-worm pair, along with the equations for the first and second limit curves. These were subsequently visualized using MATLAB R2013b.
To design the physical device, I referenced the parameters of the U10PLUS KV170 motor, which are listed in Table 1. This allowed me to determine the key design parameters for the worm gears, as shown in Table 2. I then generated the spiral lines of the worm tooth surfaces using MATLAB, output them as .ibl files, and constructed a 3D model of the entire transmission device with Creo 2.0.
| Parameter | Value |
|---|---|
| Rated Power (W) | 170 |
| Rated Voltage (V) | 24 |
| Rated Torque (Nm) | 0.54 |
| Rated Speed (rpm) | 3000 |
| Parameter | Symbol | Value |
|---|---|---|
| Center Distance | a | 100 mm |
| Base Circle Radius | r_b | 62.5 mm |
| Internal Gear Width (Design) | B | 110 mm |
| Internal Gear Width (Working) | B’ | 75 mm |
| Number of Worm Threads | z_1 | 1 |
| Number of Internal Gear Teeth | z_2 | 60 |
| Base Plane Inclination Angle | β | 12° |
| Pressure Angle | α | 22.99° |
One of the key findings in my study was that the beveloid internal gear features symmetrical wedge-shaped teeth. This geometric characteristic is particularly advantageous during the assembly process, as it allows for the adjustment of the internal gear’s relative axial position. Through this adjustment, I was able to achieve an interference-free virtual assembly of the internal gear and the worm in the Creo environment. This adjustability is a distinct advantage over conventional worm gears, as it provides a mechanism for backlash adjustment and wear compensation during the service life of the transmission.
In terms of structural compactness, the drum-worm pair designed in my research has a center distance of 100 mm. When compared to a toroidal worm pair with the same design parameters but a center distance of 220 mm, the drum-worm pair’s significantly reduced center distance demonstrates its ability to create a much more compact structure. This is crucial for applications like robotic joints, where space is at a premium. The internal gear was initially designed with a width of 110 mm, but based on the distribution of the contact lines on both the A-side and B-side tooth surfaces of the worm-wheel, I determined that the effective working width could be reduced to 75 mm. This optimization reduces material costs and manufacturing complexity.
The contact state analysis of the worm gears was a critical part of my work. The meshing equation for the A-side (forward rotation) was derived as follows:
$$ v = \left( r_b + \frac{r_2 – a}{\sin \phi_3} – \frac{v}{\tan \beta} \right) \sin \beta – \frac{H \cos \beta}{\sin \phi_3} $$
Similarly, the meshing equation for the B-side (reverse rotation) is:
$$ v’ = \left( r_b + \frac{r_2 – a}{\sin \phi_2} – \frac{v’}{\tan \beta} \right) \sin \beta – \frac{H \cos \beta}{\sin \phi_2} $$
Here, \(r_b\) is the base circle radius, \(r_2\) is the pitch circle radius of the internal gear, \(a\) is the center distance, \(\beta\) is the base plane inclination angle, \(\phi_3\) and \(\phi_2\) are working angles, and \(H\) is a constant related to the device’s geometry. The contact lines derived from these equations were analyzed to determine the effective working width of the internal gear.
To ensure the absence of undercutting, which is a common and undesirable phenomenon in gear manufacturing, I analyzed the spatial relationship between the first limit curve and the worm’s tooth root line. The equation for the first limit curve is given by:
$$ \Psi = 0 $$
Combined with the conditions for the contact surface, this yields the explicit equation for the first limit curve. My analysis showed that the first limit curve is located well inside the worm’s tooth root line. This spatial arrangement confirms that no undercutting occurs during the manufacturing or operation of the worm gears, thereby ensuring the structural integrity and accuracy of the teeth.
Furthermore, I examined the second limit curve, which serves as the envelope of the contact lines on the internal gear tooth surface. The second limit curve equation is expressed as:
$$ \Phi = 0 $$
Through my calculations, I found that the second limit curve is tangent to all contact lines in the working area. This relationship confirms that the contact pattern is both continuous and stable, which is essential for smooth power transmission and high torque capacity in the worm gears.
Integrating traditional design methodologies with the unique features of the beveloid internal gear, I proposed an innovative design for the transmission device that incorporates a drive, transmission, and support into a single, integrated structure. In terms of the drive system, the motor is housed inside the worm, creating a compact motor-worm integration. For the transmission aspect, the relative axial position of the internal gear can be easily adjusted, allowing for precise backlash adjustment and effective wear compensation without the need for complex disassembly. Finally, for the support structure, a supporting shaft is used for positioning and installation. This eliminates the need for a separate housing or box, further simplifying the device’s structure and reducing its overall weight and volume.
In summary, my design of the beveloid internal gear plane enveloping external-rotor drum-worm transmission device demonstrates several significant advantages. The symmetrical wedge-shaped teeth of the internal gear facilitate easy installation and adjustment, enabling backlash control and wear compensation, which in turn increases the utilization rate of the worm gears. By designing the internal gear based on its effective working width, I was able to lower manufacturing costs. The verification of the meshing transmission’s rationality through contact line and limit curve analysis confirms the design’s technical soundness. The final structural design scheme, which integrates the motor, transmission, and support components, presents a highly viable solution for application in robotic rotation joints. This work provides a strong theoretical and practical foundation for the development of high-performance, domestically produced robot reducers based on advanced worm gears.