1. Introduction
Non-circular gears have unique applications in various mechanical systems due to their ability to provide non-uniform transmission ratios. The design methods of non-circular gears mainly include analytical method, conversion method of tooth profile, and envelope method. Among them, the conversion method of tooth profile is widely used because of its relatively simple calculation and good applicability. However, when applied to non-circular helical gears, some problems may arise. This article focuses on improving the design of high-order non-circular helical gears using a new approach.
1.1 Background of Non-Circular Gears
Non-circular gears are different from traditional circular gears. Their pitch curves are not circular, which allows for a variable transmission ratio during the meshing process. This characteristic makes them suitable for applications where a non-uniform speed or torque transmission is required, such as in some mechanical presses, textile machinery, and printing equipment.
1.2 Significance of the Research
The research on non-circular helical gears is of great significance. Firstly, it can improve the transmission performance of mechanical systems by providing more flexible transmission ratios. Secondly, it can expand the application range of gears in various industries. However, the design and analysis of non-circular helical gears are more complex than those of circular gears, and there are still many problems to be solved.
2. Basic Theory of Non-Circular Gears
2.1 Pitch Curve Design
The pitch curve is a crucial parameter in non-circular gear design. The polar coordinate expressions of the pitch curves of the driving and driven wheels are given as follows:
| Wheel | Pitch Curve Polar Coordinate Expression |
|---|---|
| Driving Wheel | |
| Driven Wheel |
where are the orders of the driving and driven wheels respectively, are the eccentricities, and are the semi-major axis lengths. The relationship between the basic parameters of the driving and driven wheels is also determined, such as the rotation center distance .
2.2 Gear Parameter Verification
- Pressure Angle Verification: The pressure angle verification formula is used to ensure that the pressure angle during gear meshing is within a reasonable range to avoid excessive wear and ensure smooth meshing.
- Root Cutting Verification: By calculating the minimum curvature radius and comparing it with the relevant parameters, root cutting can be prevented.
- Convex-Concave Verification: This verification is to ensure that the gear profile does not have an unacceptable convex-concave shape, which may affect the meshing performance.
3. Problems in Traditional Non-Circular Helical Gear Design
3.1 Helical Line Defects
In the traditional design of non-circular helical gears using the conversion method of tooth profile, the helical line determined by calculating the curvature center and radius at each tooth position has a problem. Due to the different curvatures of the pitch curve at each tooth position, when the helical line is projected onto the plane where the pitch curve is located, it does not coincide with the pitch curve evenly. As shown in Figure 2, at the position where the pitch curve curvature is small, the helical line is biased inward, and at the position where the pitch curve curvature is large, the helical line is biased outward. This uneven bias of the helical line leads to an uneven distribution of the helical teeth, which has a significant impact on the meshing performance of non-circular helical gears.
3.2 Impact on Meshing Performance
The uneven helical teeth can cause problems such as uneven load distribution during gear meshing, which may lead to premature wear of some teeth, reduce the service life of the gears, and also affect the transmission accuracy and stability of the entire gear transmission system.
4. New Design Method Using Intersection Curve
4.1 Introduction of Involute Helix Surface
The involute helix surface has unique geometric properties. When a plane intersects the involute helix surface, the intersection curve obtained has certain characteristics. In the design of non-circular helical gears, we can use the intersection curve of the involute helix surface and the pitch curve cylinder to replace the traditional helical line.
4.2 Derivation of Intersection Curve
- Coordinate System Transformation: First, we need to establish an appropriate coordinate system. The absolute coordinate system is fixed at the rotation center of the non-circular helical gear, and the relative coordinate system is extended and fixed at the curvature center of each tooth on the end face of the non-circular helical gear.
- Calculation of Curvature Center Coordinates: The curvature center coordinates of each tooth in the absolute coordinate system are calculated. For example, for the -th tooth, the curvature center in the absolute coordinate system is given by where is the curvature radius of the -th tooth and is the inclination angle of the normal equation of the -th tooth.
- Vector Equation of Involute Helix Surface: The vector equation of the involute helix surface for the -th tooth is obtained as with its parameter equation form where is the base circle radius corresponding to the -th tooth.
- Equation of Pitch Curve Cylinder: The equation of the pitch curve cylinder is
- Calculation of Intersection Curve: By combining the equation of the involute helix surface and the equation of the pitch curve cylinder and solving them simultaneously, the intersection curve for each tooth can be obtained. The general equation of the intersection curve for all teeth is which is obtained by combining relevant equations. Although the equation of the intersection curve is complex and can only be represented by combining the pitch curve cylinder and the involute helix surface, in the actual modeling process, the intersection curve can be obtained through curve intersection operations.
5. Modeling of High-Order Non-Circular Helical Gears
5.1 Parameter Setting
- Driving Wheel Parameters: The main parameters of the driving wheel of non-circular helical gears are set as shown in Table 1.
| Name | Symbol | Value |
|–|–|–|
| Normal Module | m | 3mm |
| Number of Teeth of Driving Wheel | | 21 |
| Tooth Width | | 50mm |
| Helix Angle | | 15° (right-hand) |
| Normal Pressure Angle | | 20° |
| Normal Addendum Coefficient | h | 1 |
| Normal Clearance Coefficient | | 0.25 |
| Eccentricity of Driving Wheel | | 0.15 |
| Order of Driving Wheel | | 1 |
| End Face Module | | 2.8978mm |
| End Face Pressure Angle | | 20.6494° |
| Base Cylinder Helix Angle | | 14.0759° |
| End Face Addendum Coefficient | | 0.9659 |
| End Face Clearance Coefficient | | 0.2415 |
| Addendum | | 3mm |
| Dedendum | | 3.75mm | - Driven Wheel Parameters: The parameters of the driven wheel are determined according to the parameters of the driving wheel and the convex-concave verification. The order of the driven wheel is , the eccentricity is , the center distance is , and the number of teeth is .
5.2 Tooth Position Determination
For non-circular helical gears, due to the unequal curvature radius of the pitch curve, the meshing teeth must be paired one by one to ensure the normal operation of the gears. The position distribution of the teeth is determined according to the order of the gears. For example, when the order , the number of teeth is generally designed as an odd number, and the position of the teeth on the driving and driven wheels is determined according to certain rules.
5.3 Calculation of Curvature Radius and Center
The curvature radius and the curvature center coordinate matrix of each tooth can be calculated through relevant formulas. The curvature radii of the driving and driven wheels change periodically and symmetrically during a rotation cycle, which can be used to improve the modeling efficiency of non-circular helical gears.
5.4 Calculation of Other Parameters
In addition to the intersection curve, other parameters such as the base circle radius , the rotation angle of the lower end face involute , the rotation angle of the upper end face tooth profile based on the end face tooth profile , and the pitch of the helix line on the base cylinder $S_{1\times z…
6. The Influence of Geometric Parameters on Gear Performance
6.1 The Effect of Module on Gear Strength and Size
The module is an important parameter in gear design. A larger module generally means a thicker tooth, which can increase the strength of the gear. However, it also leads to an increase in the size of the gear. In non-circular helical gears, the choice of module needs to consider both the transmission power requirements and the space limitations of the mechanical system. For example, in a high-power transmission system with sufficient space, a larger module can be selected to ensure the strength of the gears. But in a compact mechanical device, a smaller module may be necessary, and other measures need to be taken to ensure the gear strength, such as using higher-strength materials.
6.2 The Impact of Tooth Number on Transmission Ratio and Meshing Stability
The number of teeth affects the transmission ratio and meshing stability of the gears. In non-circular gears, the relationship between the number of teeth of the driving and driven wheels and the transmission ratio is more complex than that of circular gears. An appropriate choice of the number of teeth can ensure a desired non-uniform transmission ratio. At the same time, the number of teeth also affects the meshing stability. A proper number of teeth can make the gears mesh more smoothly and evenly distribute the load during meshing. For example, in some applications where a large transmission ratio change is required, a specific combination of the number of teeth of the driving and driven wheels needs to be carefully designed to achieve the best transmission performance.
6.3 The Role of Helix Angle in Load Distribution and Noise Reduction
The helix angle plays a crucial role in load distribution and noise reduction. A larger helix angle can increase the contact ratio of the gears, which means more teeth are in contact during meshing, resulting in a more even load distribution. This can reduce the stress concentration on individual teeth and increase the service life of the gears. In addition, an appropriate helix angle can also help reduce the noise generated during gear meshing. The helical teeth can gradually engage and disengage during rotation, which reduces the impact and vibration, thereby reducing the noise level. However, a too-large helix angle may also bring some problems, such as increasing the axial force on the gears, which requires appropriate bearing design to handle.
7. Manufacturing Considerations for Non-Circular Helical Gears
7.1 Machining Methods and Their Advantages and Disadvantages
- Milling Method: Milling is a common method for machining gears. For non-circular helical gears, milling can be used to shape the teeth. The advantage of milling is its versatility. It can be used to produce gears with different geometries. However, the milling process may require multiple passes and precise tool paths, which can be time-consuming and require high machining accuracy.
- ** Hobbing Method**: Hobbing is another widely used gear machining method. It has a high production efficiency and can produce gears with good quality. For non-circular helical gears, hobbing can be used if the pitch curve and helix angle are properly designed. The disadvantage of hobbing is that it may not be suitable for gears with very complex geometries, and the tool setup and adjustment may be more complicated.
- Additive Manufacturing Method: Additive manufacturing, such as 3D printing, has emerged as a new option for gear manufacturing. It allows for the production of complex geometries with relative ease. For non-circular helical gears, 3D printing can be used to produce prototypes or small batches of gears. However, the mechanical properties of 3D-printed gears may not be as good as those of traditionally machined gears, and further post-processing may be required to improve their quality.
7.2 Tool Selection and Its Influence on Gear Quality
The selection of tools is crucial for the manufacturing quality of non-circular helical gears. Different machining methods require different tools. For example, in milling, the selection of the milling cutter depends on the module, helix angle, and tooth profile of the gear. A proper milling cutter can ensure the accuracy of the tooth profile and the surface finish of the gear. In hobbing, the hob cutter needs to be selected according to the pitch curve and other parameters of the gear. The quality of the hob cutter directly affects the quality of the machined gear. In additive manufacturing, the nozzle or laser settings can be considered as a kind of “tool” selection, which affects the resolution and material deposition of the printed part, and thus the quality of the gear.
7.3 Tolerance Control and Its Importance in Gear Manufacturing
Tolerance control is essential in gear manufacturing. In non-circular helical gears, tight tolerance control is required to ensure proper meshing and transmission performance. Tolerances need to be set for various parameters such as the pitch curve, tooth profile, and helix angle. If the tolerances are too loose, it may lead to problems such as improper meshing, increased noise, and reduced transmission efficiency. On the other hand, if the tolerances are too tight, it may increase the manufacturing cost and difficulty. Therefore, a balance needs to be struck between tolerance requirements and manufacturing feasibility.
8. Application Examples of Non-Circular Helical Gears
8.1 Use in Mechanical Presses
In mechanical presses, non-circular helical gears can be used to provide a non-uniform motion of the press ram. The variable transmission ratio of the non-circular helical gears allows for a controlled acceleration and deceleration of the ram during the stamping process. This can improve the quality of the stamped parts by ensuring a more uniform force distribution during the stamping operation. For example, in a deep drawing process, the non-uniform motion of the ram provided by the non-circular helical gears can help prevent wrinkling and tearing of the sheet metal.
8.2 Application in Textile Machinery
In textile machinery, non-circular helical gears can be used to control the speed and tension of the yarn. The non-uniform transmission ratio of the gears can be adjusted according to the different stages of the textile process. For example, during the winding process, a specific transmission ratio can be set to ensure a proper tension of the yarn on the bobbin. During the unwinding process, a different transmission ratio may be required to control the speed of the yarn supply. This helps to improve the quality and efficiency of the textile production process.
8.3 Employment in Printing Equipment
In printing equipment, non-circular helical gears can be used to control the movement of the printing rollers. The variable transmission ratio allows for a more precise control of the roller speed and position, which is crucial for accurate printing. For example, in a color printing process, different transmission ratios may be required for different color rollers to ensure that the colors are accurately overlaid and the printed image has high quality.
9. Comparison with Other Gear Types
9.1 Non-Circular Helical Gears vs. Circular Gears
- Transmission Ratio: Circular gears have a constant transmission ratio, while non-circular helical gears can provide a variable transmission ratio. This makes non-circular helical gears more suitable for applications where a non-uniform speed or torque transmission is required.
- Meshing Characteristics: The meshing of circular gears is relatively simple and uniform, while the meshing of non-circular helical gears is more complex due to the non-uniform pitch curve and helix angle. However, the complex meshing of non-circular helical gears can also provide better load distribution and transmission performance in some cases.
- Application Range: Circular gears are widely used in many general mechanical systems, while non-circular helical gears are mainly used in specialized applications where a non-uniform transmission is needed.
9.2 Non-Circular Helical Gears vs. Non-Circular Spur Gears
- Helix Angle and Load Distribution: Non-circular helical gears have a helix angle, which can provide better load distribution compared to non-circular spur gears. The helical teeth of non-circular helical gears can engage and disengage more gradually, reducing the impact and stress concentration during meshing.
- Transmission Efficiency and Noise: Due to the better load distribution and more gradual meshing, non-circular helical gears generally have higher transmission efficiency and lower noise levels compared to non-circular spur gears.
- Manufacturing Complexity: The manufacturing of non-circular helical gears is more complex than that of non-circular spur gears because of the need to consider the helix angle and its associated machining and design requirements.
10. Challenges and Opportunities in the Development of Non-Circular Helical Gears
10.1 Technical Challenges
- Design Complexity: The design of non-circular helical gears is more complex than that of traditional circular gears. The non-uniform pitch curve and helix angle require more advanced design methods and tools. There is a need for further research on how to optimize the design to ensure better performance and reliability.
- Manufacturing Difficulty: The manufacturing of non-circular helical gears is also challenging. The complex geometry requires precise machining and tight tolerance control. There is a need for improved manufacturing techniques and equipment to meet the quality requirements of non-circular helical gears.
- Analysis and Simulation Difficulty: Analyzing and simulating the performance of non-circular helical gears is difficult due to their complex geometry and variable transmission ratio. There is a need for more accurate analysis and simulation methods to predict the behavior of the gears during operation.
10.1 Opportunities
- Increasing Application Demand: With the development of various industries, there is an increasing demand for non-uniform transmission in mechanical systems. Non-circular helical gears can meet this demand and have potential applications in many fields such as automotive, aerospace, and robotics.
- Advancement in Technology: The continuous advancement in computer-aided design (CAD), computer-aided manufacturing (CAM), and simulation technology provides opportunities for the development of non-circular helical gears. These technologies can be used to improve the design, manufacturing, and analysis of non-circular helical gears.
- Material Innovation: The development of new materials with better mechanical properties can also improve the performance of non-circular helical gears. New materials can be used to increase the strength, durability, and wear resistance of the gears.
11. Conclusion
11.1 Recapitulation of Key Points
This article has comprehensively discussed the design, analysis, manufacturing, and application of non-circular helical gears. The new design method using the intersection curve of the involute helix surface and the pitch curve cylinder has been introduced to address the problems in traditional non-circular helical gear design. The influence of geometric parameters on gear performance, manufacturing considerations, application examples, and comparisons with other gear types have also been explored. The challenges and opportunities in the development of non-circular helical gears have been analyzed.
11.2 Implications for Future Research and Practice
Future research should focus on further optimizing the design method of non-circular helical gears to improve their performance and reliability. The manufacturing techniques need to be continuously improved to meet the quality requirements of non-circular helical gears. The analysis and simulation methods should be enhanced to accurately predict the behavior of the gears during operation. In practice, the application range of non-circular helical gears should be further expanded to meet the increasing demand for non-uniform transmission in various industries.