Over the past four decades, planar double-enveloping hourglass worm gears have been extensively studied and applied in China. Compared with cylindrical worm gears, this type of worm gears exhibits superior load capacity, higher transmission efficiency, and longer service life due to its multi-tooth engagement, double-line contact, and large lubrication angle. However, the design of parameters, accuracy control, and manufacturing cost remain challenging. In this review, we summarize the research and development status of planar double-enveloping hourglass worm gears from three perspectives: design, manufacturing, and measurement. We also propose future research directions to promote the high-efficiency and high-precision manufacturing of these worm gears.
1. Design Technology
1.1 3D Modeling
The tooth surfaces of planar enveloping hourglass worm and the double-enveloping worm wheel are spatially complex. We have developed two main approaches for 3D modeling. The first approach is based on the meshing principle: we establish tooth surface equations using software like MATLAB, Visual C++, and VB, then import point clouds into CAD software to fit surfaces. This method is time-consuming and the deviation between the model and the theoretical surface must be checked. The second approach is direct digital modeling: we simulate the actual cutting process using Boolean operations. The worm and worm wheel are rotated step by step; the step size determines model accuracy and data volume. This method avoids solving complex meshing equations but becomes complicated when considering deformation and modification.
1.2 Meshing Characteristic Analysis
Before the 1990s, we had already obtained the instantaneous contact lines, contact areas, boundary curves, induced curvature radii, relative entrainment velocities, and lubrication angles for unmodified worm gears. For example, for a center distance of 125 mm and transmission ratio of 40, the worm wheel tooth surface shows crossing contact lines in certain zones, which are prone to fatigue pitting. In recent years, we have focused on three aspects:
- Effect of parameters (inclination angle of the mother plane, main base circle diameter, module, number of teeth of the tool gear) on contact state and meshing performance.
- Load distribution: we found that the contact load decreases from the entry to the exit of engagement. The maximum load occurs near the second boundary curve in the secondary contact region. Contact stress along the contact line shows an “L” or “U” shape, confirmed by photoelastic freezing.
- Tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA): we analyzed the influence of manufacturing errors and elastic deformation on the contact state and transmission accuracy, and calculated the contact stress and load sharing under actual working conditions.
1.3 Parameter Optimization
Under fixed center distance and transmission ratio, we have proposed eight different optimization methods, differing in design variables, constraints, objectives, and algorithms. The optimization models are summarized in Table 1.
| Component | Content |
|---|---|
| Design variables | Mother plane inclination, main base circle diameter, pitch circle diameter, number of worm threads |
| Objectives | Minimize distance from first contact line intersection to worm wheel midline, minimize relative sliding speed and wheel size, maximize minimum oil film thickness, maximize efficiency, minimize cost, or a weighted combination |
| Constraints | Geometric (parameters in feasible range), meshing performance (no undercut, proper starting angle, reasonable contact line distribution), strength (worm bending, wheel contact and bending strength) |
| Algorithms | Complex method, multi-objective programming, exterior penalty function, fuzzy algorithm |
We have also developed design software, such as the “Huanguang Software” (环光软件), which has been tested for 20 years. It includes parameter optimization, hob design, and efficiency and power calculation. However, many manufacturers still rely on Excel spreadsheets with manual formulas, lacking effective evaluation of macroscopic and microscopic meshing performance.
1.4 Modification Design
Modification improves worm gear performance. Two types exist:
1.4.1 Modification Based on Meshing Principle
This involves making the relative motion parameters of the first enveloping process unequal to those of the second process (e.g., center distance, transmission ratio, axial position). The worm and hob are modified identically, resulting in line contact. According to the presence of the second boundary curve on the worm wheel, we classify modified drives into Type I and Type II, with some arguing for a Type III. The comparison is shown in Table 2.
| Type | Advantages | Disadvantages |
|---|---|---|
| Unmodified | Full tooth contact on both worm and wheel | Crossing contact lines on wheel |
| Type I | Full contact on wheel, no crossing lines | Short contact zone on worm, second boundary curve, arched contact lines with poor lubrication at crest |
| Type II | Full contact on worm, no crossing lines | Post-transition zone and first boundary curve on wheel |
Some authors recommend Type I, others Type II. The choice of modification amount is critical. For example, if the center distance modification is too small, the first boundary curve may not be removed. We can combine center distance and transmission ratio modifications to enable larger values.
1.4.2 Mismatched Modification
This method reduces the sensitivity of worm gears to errors and deformation. We have proposed modifying the tooth height direction to control contact area and the helical direction to reduce transmission errors caused by errors and load.
2. Manufacturing Technology
The manufacturing of planar enveloping hourglass worm and double-enveloping worm wheel directly affects transmission quality. Traditional machining uses dedicated machines with a rotary table, but they are complex and expensive to adjust. New CNC technologies have emerged.
2.1 CNC Machining Technologies
2.1.1 CNC Turning of Worm
We mount the worm blank on a CNC lathe and use three-axis simultaneous motion to control the turning tool along the tooth surface calculated from the meshing principle. This is slow and low-accuracy. Recently, we developed a method using a standard cutting tool and macro programs to achieve variable-lead rough turning on conventional lathes, which significantly improves efficiency. However, the resulting profile is a straight-sided hourglass worm that requires further grinding, and the grinding allowance is uneven: more at the entry and less at the exit, and more at the tip than the root.
2.1.2 Virtual Rotary Center Grinding
This is used for precision grinding of the worm. The machine has a lathe-like bed with a double-layer worktable carrying a rotary table and a grinding head. The worktable performs circular interpolation in X and Z directions, while the rotary table rotates (B-axis) and the worm rotates (C-axis). The relationship is: the rotation angle of the circular interpolation equals the rotation of the rotary table, and the worm rotation angle equals the rotary table angle multiplied by the transmission ratio. This eliminates the need for center distance and base circle adjustments, and the rotary table size is reduced, expanding the machining range. We can also replace the grinding head with a turning tool or milling cutter for roughing.
2.1.3 Five-Axis Flank Milling
The worm tooth surface is a developable ruled surface, so we can perform flank milling on a five-axis machining center. The tool path is generated from instantaneous contact lines. Roughing paths are offset by the allowance, and finishing paths make the tool side edge coincide with the contact line. This solves the uneven allowance problem and improves efficiency and accuracy.
2.1.4 CNC Milling of Worm Wheel
For large, single-piece, or small-batch worm wheels, we avoid expensive hobs and use CNC milling. We first create a 3D model, then generate tool paths. We have proposed using a ball-end cutter with three-axis motion, discretizing the tooth space with section planes, and approximating the theoretical surface with Bezier patches. Four-axis CNC machining has also been validated through simulation.
2.1.5 CNC Relief Grinding of Hob
The relief grinding of the hob for double-enveloping worm wheels is one of the most difficult tasks. We proposed a CNC method using a four-axis grinding machine with virtual rotary center technology. By controlling the relative positions of four axes, the normal vector of the grinding plane is kept parallel to the normal of the tooth flank at the cutting edge, the grinding plane always contacts the cutting edge point, and the edge of the grinding wheel always touches the hob root to prevent interference. The resulting hob has a constant relief angle (see Figure 6 in original).
2.2 Comparison of Manufacturing Technologies
We summarize the main manufacturing methods for worm and worm wheel in Tables 3 and 4.
| Method | Advantages | Disadvantages | Application |
|---|---|---|---|
| Traditional dedicated machine | Mature technology | High cost, long cycle, single-purpose | General |
| CNC turning | High efficiency, general-purpose machine | Low accuracy | Roughing |
| Virtual rotary center grinding | High efficiency and accuracy | High cost, dedicated machine | Batch finishing |
| Five-axis flank milling | High accuracy, general-purpose machine | Even higher cost, low efficiency | Single-piece finishing |
| Method | Advantages | Disadvantages | Application |
|---|---|---|---|
| Fly cutter | Simple tool, low cost | Poor enveloping effect | Small batch, low precision |
| Hob | High precision, high efficiency | Difficult to manufacture, high cost | Best overall productivity |
| Shaving cutter | Higher precision | Extremely difficult to make, very high cost | High precision requirements |
| Lapping wheel | Saves running-in time | Very low efficiency | High precision |
| Milling cutter | Standard tool, accurate forming | High machine requirement | Small batch |
3. Measurement Technology
Measurement of worm and hob errors has been a bottleneck. We have developed coordinate measurement methods using CMMs, gear measuring centers, or dedicated instruments. Compared to CMMs, gear measuring centers are more efficient.
3.1 Measurement of Worm and Hob Errors
We designed a dedicated hourglass worm and hob measuring instrument (see Figure 7 in original). It is a horizontal structure with three linear axes (X, Z, Y) and one rotary axis (θ). It works on the electronic generating principle. The probe moves along X and Z while the worm rotates, and the data from scales and probe are processed to obtain helix errors, tooth profile errors at the throat section, indexing errors, and tooth surface topography. It can also measure ZA, ZN, ZI cylindrical worms and their hobs. This instrument has been used in production for over three years, effectively solving the accuracy measurement problem.
3.2 Measurement of Worm Gear Pair Errors
In the 1990s, we developed a double-enveloping worm gear pair tester using a plane probe to measure tangential composite errors. Recently, a new testing device was introduced: it consists of a spindle, tailstock, rotary table, and three slides. The worm wheel is mounted on the rotary table, and the worm is mounted between spindle and tailstock. Angular encoders on the spindle and rotary table measure the rotational relationship to derive tangential composite error and single-flank composite error. The radial slide measures center distance error, and contact pattern is assessed by coating with red lead.
4. Future Research Directions
Despite progress, systematic process specifications are still incomplete. Design and manufacturing often rely on experience and analogy, lacking reliable detection and verification, leading to high cost, low precision, and poor quality. We identify six key areas for further investigation.
4.1 Modification Theory
Current studies focus on instantaneous effects without considering the entire meshing cycle or load. We need to analyze how different modification amounts remove the first boundary curve, determine the post-transition zone, and calculate the shortest contact line length during the cycle (the weakest point for contact fatigue). Finite element simulation should be used to compare macroscopic contact patterns and microscopic performance under different modifications, enabling better parameter optimization.
4.2 Manufacturing Error Correction Technology
With dedicated measuring instruments now available, we need to use measurement data to correct errors. The machining of worm gears involves multiple process parameters and many coupled axes. By establishing a homogeneous coordinate transformation matrix containing assumed errors, we can build a mathematical model of the worm with manufacturing errors. Error simulation reveals the mapping between individual errors and tooth surface deviations, guiding production. We aim to construct a closed-loop manufacturing system integrating design, manufacturing, measurement, and error compensation for worms and hobs. Similar work is needed for worm wheel measurement and correction.
4.3 Assembly Technology for Worm Gear Pairs
After manufacturing, assembly adjustments based on contact pattern are done empirically. We need to use TCA under combined manufacturing and installation errors to predict contact patterns and transmission error curves. This will guide assembly and improve performance. A closed-loop assembly system should be built.
4.4 Hob Manufacturing Technology
When the grinding wheel radius is large relative to the hob, interference may occur in relief grinding, especially with many gashes or large helix angles. We need to modify the machine or adopt new techniques to grind the rake face, primary flank, and secondary flank in one setup. For assembled hobs, pre-forming the inserts can avoid later relief grinding.
4.5 Improvement of Technical Standards
The accuracy standard for planar double-enveloping hourglass worm gears was issued in 1996 and is now outdated. We need to update it based on new measurement equipment and methods. Furthermore, design parameters are not unified; different manufacturers produce non-interchangeable parts. Standardization of parameters will improve interchangeability. Standards for worm wheels, hobs, and new machining equipment should also be developed.
4.6 Development of Industrial Software
Design and manufacturing still rely on experience. We need mature software that integrates parameter optimization, strength check, macroscopic and microscopic meshing analysis, 3D modeling, and even grinding allowance homogenization. Manufacturing process software should generate NC programs automatically and connect with measurement software to modify process parameters. Open collaboration among researchers is essential to gather data and improve software continuously.
5. Conclusion
Planar double-enveloping hourglass worm gears have unique advantages, but their widespread application is hindered by difficulties in design, manufacturing, and measurement. With the development of new CNC technologies and measuring instruments, we have made significant progress. However, challenges remain in modification theory, error correction, assembly, hob manufacturing, standards, and software. By solving these problems and establishing a closed-loop system integrating design, manufacturing, and testing, we can achieve high-efficiency and high-precision manufacturing, further promoting the development of these advanced worm gears.