This article focuses on the research of spiral bevel gears with a small number of teeth. It first introduces the background and significance of the research, then details the design methods of geometric parameters, the derivation of tooth surface equations, the solution of machining parameters, the simulation of tooth surface meshing, and the gear-cutting and rolling inspection tests. Through a series of theoretical derivations and experimental verifications, it proves the feasibility of the geometric design and machining methods of high reduction ratio spiral bevel gears, providing a theoretical basis and practical reference for the design and manufacturing of such gears.
1. Introduction
In modern mechanical transmission systems, the demand for high-power density, high integration, and high efficiency is increasing. Spiral bevel gears play a crucial role in transmitting power between intersecting shafts. Among them, gears with a small number of teeth and high reduction ratios have the advantages of small size and light weight, which are widely used in various fields such as aerospace, automotive, and robotics. However, the design and manufacturing of such gears face many challenges due to their special geometric characteristics and meshing requirements.
2. Geometric Design of Spiral Bevel Gears
2.1 Design Principles
The design of spiral bevel gears in this article is based on the principles of comprehensive displacement and pitch cone external meshing. This design method can balance the strength of the large and small gears and improve the load-carrying capacity of the gear pair. At the same time, pitch cone external meshing can also adjust the tooth surface sliding and prevent the friction force from reversing.
2.2 Geometric Parameters and Constraints
A pair of arc contoured bevel gears with a reduction ratio of 4:41 is designed as an example. The geometric parameters include the number of teeth, module, pressure angle, shaft angle, and tooth width. The following constraints need to be considered during the design process:
3. Tooth Surface Equation Derivation
3.1 Large Gear Tooth Surface Equation
The large gear is processed by the forming method. The tool disc equation of the large gear is a rotating conical surface. After coordinate transformation, the tooth surface equation of the large gear in the large gear coordinate system can be obtained.
3.2 Conjugate Small Gear Tooth Surface Equation
The conjugate small gear tooth surface equation is derived based on the conjugate meshing relationship with the large gear. The meshing relationship between the large and small gears is analyzed, and the tooth surface radius vector and normal vector of the conjugate small gear are obtained through the meshing equation.
3.3 Small Gear Processing Method and Tooth Surface Equation
The small gear is processed by the single-sided generating method, and the tool disc of the small gear adopts circular edge modification. The tool disc equation and the tooth surface equation of the small gear in the small gear coordinate system are derived.
4. Machining Parameter Solution of the Small Gear Based on the Curved-Surface Synthesis Method
4.1 Curved-Surface Synthesis Contact Parameter Calculation
The conjugate small gear tooth surface, the modified tooth surface, and the difference surface (ease-of) are regarded as three closely tangent surfaces. The normal comprehensive error between the conjugate small gear tooth surface and the modified tooth surface reflects the tooth surface modification gradient. The ease-of surface is represented by a quadratic paraboloid, and its main curvatures determine the size, shape, and direction of the tooth surface contact area and the transmission error.
4.2 Small Gear Machining Parameter Solution
By establishing equations based on the condition that the same point on the same surface should satisfy certain conditions, a system of equations consisting of 9 independent equations is obtained. These equations include small gear machining parameters, small gear tool disc parameters, and curve coordinate parameters. By using a nonlinear constraint optimization solution method, the machining parameters of the small gear are calculated.
| Parameter Name | Convex Surface | Concave Surface |
|---|---|---|
| Tool Pressure Angle /() | -16.0 | 24.0 |
| Radial Tool Position /mm | 42.9147 | 46.7355 |
| Angular Tool Position /() | 65.2822 | 64.6142 |
| Wheel Blank Installation Angle /() | 5.5722 | 5.5722 |
| Bed Position /mm | -0.4497 | -0.5839 |
| Axial Wheel Position /mm | -0.9876 | 0.3946 |
| Roll Ratio | 10.6656 | 9.9004 |
| Tool Tip Radius /mm | 58.5734 | 55.3288 |
| Blade Circular Arc Radius /mm | 32.0 | 80.0 |
| Vertical Wheel Position /mm | 0.0 | 0.0 |
5. Tooth Surface Meshing Simulation
5.1 Ease-of Difference Surface
The difference surface between the conjugate small gear tooth surface and the modified tooth surface is analyzed. The difference tooth surfaces of the convex and concave surfaces of the small gear are shown, and it can be seen that they meet the requirements of the tooth surface modification gradient.
5.2 Contact Path and Transmission Error Simulation
The contact path and contact lines on the tooth surface are simulated. The contact paths of the convex and concave surfaces of the small gear are from the root of the large end to the tip of the small end and from the root of the small end to the tip of the large end, respectively, which are internal diagonal contacts and meet the design requirements. The transmission error curve is obtained, and it is parabolic, which also meets the design requirements.
5.3 Instantaneous Contact Simulation
Based on the machining parameters of the small gear, a three-dimensional model of the small gear is established and assembled with the large gear for dynamic simulation in UG. The instantaneous contact state of the gear pair, the contact path from the entry to the exit of the meshing of the teeth, and the instantaneous contact situation can be observed, and the instantaneous contact area is an ideal ellipse.
6. Gear-Cutting and Rolling Inspection Tests
6.1 Gear-Cutting Test
The gear-cutting test is carried out on the GH-35 machine tool. The machining parameters are converted into the adjustment parameters of the GH-35 machine tool according to the different processing methods of the large and small gears. The cutting process of the small gear is shown.
6.2 Rolling Inspection Test
After the gear-cutting test, the rolling inspection of the large and small gears is carried out. It can be observed that there is no distortion at the tooth tips of the large and small gears. The contact spots on the tooth surface obtained after the final rolling inspection are elliptical.
7. Illustrations
Figure 1: Large Gear Tool Disc Coordinate System
This figure shows the coordinate system of the large gear tool disc. The radius vector and normal vector of the tool disc are defined in this coordinate system, which is crucial for deriving the tooth surface equation of the large gear.
Figure 2: Machining Parameters of the Large Gear
It illustrates the machining parameters of the large gear when processed by the forming method. These parameters include radial tool position, angular tool position, bed position, and axial wheel position correction value.
Figure 3: Meshing Relationship between Large and Small Gears
The figure depicts the meshing relationship between the large and small gears. It shows the relative positions and rotations of the two gears in different coordinate systems, providing a basis for deriving the conjugate small gear tooth surface equation.
Figure 4: Small Gear Tool Disc Coordinate System
Here, the coordinate system of the small gear tool disc with circular edge modification is presented. The radius vector and normal vector of the tool disc in this system are used to derive the tooth surface equation of the small gear.
Figure 5: Small Gear Machining Coordinate System
This figure shows the machining coordinate system of the small gear. It includes the machine tool fixed coordinate system, the generating wheel coordinate system, and the small gear coordinate system, along with the relationships between them and the determination of various positions and angles.
Figure 6: Tooth Surface Modification Gradient and Transmission Error
It visually represents the relationship between the tooth surface modification gradient and the transmission error. The ease-of surface is used to analyze how the modification affects the contact area and transmission characteristics of the tooth surface.
Figure 7: Ease-of Surface and Its Parameters
The figure shows the ease-of surface and its related parameters such as the main curvatures and the shape of the surface. These parameters play a significant role in determining the tooth surface contact behavior.
Figure 8: Small Gear Convex and Concave Surface Difference Tooth Surfaces
This shows the difference tooth surfaces of the convex and concave surfaces of the small gear. It helps to understand the deviation between the conjugate tooth surface and the modified tooth surface and verify if it meets the modification gradient requirements.
Figure 9: Small Gear Tooth Surface Contact Path and Contact Lines Simulation Results
The simulation results of the contact path and contact lines on the small gear tooth surface are presented. It shows the internal diagonal contact paths of the convex and concave surfaces, which is an important indicator of the correct meshing design.
Figure 10: Transmission Error Curve
The transmission error curve with the small gear rotation angle as the abscissa and the deviation as the ordinate is depicted. The parabolic shape of the curve indicates that the transmission error meets the design requirements.
Figure 11: Gear Pair Instantaneous Contact Ellipse
This figure shows the instantaneous contact ellipse of the gear pair during dynamic simulation. It reflects the actual contact state between the gears during meshing.
Figure 12: Small Gear Cutting Process
It illustrates the cutting process of the small gear during the gear-cutting test, showing how the tool interacts with the gear blank to form the teeth.
Figure 13: Large and Small Gear Rolling Inspection
This shows the process of rolling inspection of the large and small gears. It helps to observe the meshing and contact conditions between the gears after cutting.
Figure 14: Large Gear Contact Imprint
The contact imprint on the large gear after rolling inspection is presented. The elliptical shape and the internal diagonal contact without tooth tip contact verify the good performance of the gear meshing.
8. Significance and Applications
The research on the machining parameters calculation simulation and gear-cutting test of spiral bevel gears with a small number of teeth based on the curved – surface synthesis method has several important implications and applications.
8.1 Significance in Gear Design Theory
- Innovative Design Method: The proposed comprehensive displacement and pitch cone external meshing design principles offer a new approach for the geometric design of spiral bevel gears. This method takes into account the balance of strength between the large and small gears and the control of tooth surface sliding, which is beneficial for improving the overall performance and reliability of the gear pair.
- Enhanced Understanding of Tooth Surface Modification: By using the curved – surface synthesis method to analyze the tooth surface modification gradient and its relationship with transmission error and contact characteristics, a deeper understanding of the tooth surface modification mechanism is achieved. This knowledge can be used to optimize the design of tooth surface modification in future gear designs.
8.2 Applications in Engineering Fields
- Automotive Industry: In automotive transmissions, the use of spiral bevel gears with a small number of teeth and high reduction ratios can help to reduce the size and weight of the transmission system, thereby improving fuel efficiency and vehicle performance. The research results can be directly applied to the design and manufacturing of automotive gears to ensure their proper meshing and reliable operation.
- Aerospace Engineering: In aerospace applications, where weight and space are critical factors, the small and lightweight spiral bevel gears designed using this method can be used in various power transmission systems. The accurate calculation of machining parameters and simulation of tooth surface meshing ensure the high precision and reliability required in aerospace gears.
- Robotics: In robotic joints, the need for compact and efficient power transmission is increasing. The spiral bevel gears studied in this article can meet these requirements and provide a smooth and reliable power transmission solution for robotic systems.
9. Future Research Directions
Although the current research has achieved certain results in the design and manufacturing of spiral bevel gears with a small number of teeth, there are still several areas that can be further explored and improved.
9.1 Improvement of Gear Manufacturing Accuracy
- Advanced Machining Technologies: The development and application of more advanced machining technologies, such as five – axis machining and precision grinding, can further improve the accuracy of gear manufacturing. These technologies can reduce machining errors and improve the surface quality of the gears, thereby enhancing their meshing performance and load – carrying capacity.
- Error Compensation Techniques: Research on error compensation techniques during gear manufacturing is also necessary. By accurately measuring and analyzing machining errors and then applying appropriate compensation methods, the accuracy of the final gears can be significantly improved. This includes techniques for compensating for geometric errors, thermal errors, and cutting force – induced errors.
9.2 Optimization of Tooth Surface Modification Design
- Advanced Modification Algorithms: The development of more advanced tooth surface modification algorithms is needed. These algorithms should be able to consider more factors such as the actual operating conditions of the gears, the load distribution on the tooth surface, and the dynamic characteristics of the meshing process. By using these algorithms, a more optimized tooth surface modification design can be achieved to further improve the performance of the gears.
- Multi – objective Optimization: In tooth surface modification design, multi – objective optimization considering multiple performance indicators such as contact stress, transmission error, and tooth surface wear should be carried out. This requires the development of appropriate optimization models and algorithms to balance the different performance requirements and achieve an overall optimal design.
9.3 Exploration of New Materials and Manufacturing Technologies in Gear Manufacturing
- High – Performance Materials: The exploration and application of high – performance materials in gear manufacturing can significantly improve the mechanical properties of the gears. For example, the use of advanced composites or superalloys can increase the strength and hardness of the gears while reducing their weight. Research on the compatibility of these materials with existing manufacturing processes and the optimization of their processing parameters is essential.
- Additive Manufacturing: The application of additive manufacturing technologies, such as 3D printing, in gear manufacturing is an emerging trend. This technology can offer unique advantages such as the ability to produce complex geometries and customized designs. However, further research is needed to address issues such as the accuracy and mechanical properties of printed gears and the optimization of the printing process.
In conclusion, the research on spiral bevel gears with a small number of teeth has made important progress, but there is still much room for development. Future research should focus on improving manufacturing accuracy, optimizing tooth surface modification design, and exploring new materials and manufacturing technologies to meet the increasing demands of modern mechanical transmission systems.