1. Introduction
Spiral bevel gears (Gleason system) play a crucial role in various mechanical systems, especially in high – speed and heavy – load applications such as automobiles, construction machinery, aerospace, and ships. Their high – efficiency power transmission, compact structure, and stable operation make them indispensable components. However, due to the complex tooth surface geometry of spiral bevel gears, accurately modeling and analyzing their contact behavior during meshing is a challenging task. This article aims to comprehensively introduce the parameter modeling and finite – element contact analysis of spiral bevel gears, providing a detailed guide for engineers and researchers in this field.
1.1 Significance of Spiral Bevel Gears
Spiral bevel gears are widely used because of their unique advantages. Table 1 summarizes the main characteristics and application scenarios of spiral bevel gears.
| Characteristics | Description | Application Scenarios |
|---|---|---|
| High – efficiency power transmission | They can transfer power with relatively low energy loss, which is crucial for energy – saving mechanical systems. | Automobile transmissions, where high – efficiency power transfer is required to improve fuel economy. |
| Compact structure | Occupy less space, making them suitable for applications with limited installation space. | Small – sized construction machinery and precision instruments. |
| Stable operation | Provide smooth and stable power transmission, reducing vibration and noise. | Aerospace equipment, where vibration and noise control are of great importance for the safety and comfort of passengers and the normal operation of equipment. |
1.2 Research Background and Motivation
The accurate analysis of the tooth – surface contact behavior of spiral bevel gears during meshing is essential for improving their performance and reliability. In the past, due to the complexity of the tooth – surface geometry, it was difficult to establish a precise model to describe the actual contact situation. However, with the development of computer – aided design (CAD) and finite – element analysis (FEA) technologies, it has become possible to accurately simulate and analyze the contact process of spiral bevel gears. The research on spiral bevel gear modeling and contact analysis has attracted the attention of many scholars at home and abroad. Different research methods and achievements have been proposed, as shown in Table 2.
| Researcher(s) | Research Methods | Achievements |
|---|---|---|
| Hou Xiangying et al. | Transformed theoretical calculation formulas into numerical algorithms, and analyzed tooth – surface contact changes through finite – element node coordinate solving and mesh element division. | Provided a numerical method for analyzing tooth – surface contact changes. |
| Litvin | Expounded the geometry and meshing principle of spiral bevel gears, and proposed the theory of using computer simulation and coordinate transformation for gear tooth formation. | Laid a theoretical foundation for gear – tooth – surface modeling. |
| Yang Bohui et al. | Calculated tooth – surface grid – node coordinates accurately through tooth – surface equations, and generated a precise finite – element model of spiral bevel gear meshing transmission. | Improved the accuracy of finite – element model generation. |
| Wang Yongsheng et al. | Solved the parametric equation expressions of the curves forming the tooth profile of the gear’s large and small end faces and the tooth line, and imported them into software to generate a tooth – surface model. | Achieved parametric modeling of the tooth – surface. |
| Fang Zongde et al. | Considered the edge – contact of spiral bevel gears for load – carrying contact analysis and proposed a geometric analysis method for edge – contact. | Provided a new method for load – carrying meshing simulation. |
| Mou Yanming et al. | Studied the impact of load and speed on the meshing tooth – surface based on the meshing – transmission characteristics of spiral bevel gears, combined with tooth – surface contact analysis and load – carrying contact analysis. | Analyzed the influencing factors of meshing – tooth – surface impact. |
2. Tooth – Surface Modeling of Spiral Bevel Gears
2.1 Tooth – Surface Equation of the Large Gear
The right – hand large gear of the spiral bevel gear is usually processed by the generating method. The tooth surface is a part of the envelope surface of the cutting – tool trajectory. During the processing, the cutter head rotates around its axis to form a cutting cone surface, and the cradle and the large – gear blank also rotate around their axes to form the large – gear tooth surface.
To describe the position of the large – gear cutter head and the gear blank, coordinate systems are established. As shown in Figure 1, is the machine – tool center, is the cutter – head center, the plane is the plane where the cutting – tool tip is located (coinciding with the machine – tool plane ), is the axial cutter position, and is the angular cutter position. As shown in Figure 2, is the design cone vertex of the gear blank, is the bed position, is the axial gear position, and is the gear – blank installation angle.
4.1.2 Meshing
The accuracy and type of meshing directly affect the accuracy of the three – dimensional model analysis results. If the mesh is too dense, it will increase the calculation time and occupy a large amount of computer resources. To obtain accurate calculation results and improve calculation efficiency, tetrahedral meshing is selected to model the gears. The expected meshing gear contact surfaces are locally refined. The refined part has a mesh size of 1 mm, and the non – refined part has a mesh size of 10 mm. The finite – element mesh model is shown in Figure 9. There are a total of 619,607 nodes and 418,275 elements.
4.1.3 Setting Boundary Conditions and Contact Relationships
During the operation of the spiral bevel gear pair, the motor applies a torque that rotates around the central axis to the driving gear through the shaft. The driving gear and the driven gear mesh to drive the driven gear to rotate around the central axis. The driven gear has not only an input torque but also a load. Therefore, the applied load is that the driving gear rotates around the central axis at a speed of 600 r/min. To compare the influence of the load on the contact area, the resistance torques of the driven gear are set to 30 N·m and 500 N·m respectively.
The locally refined tooth surfaces are selected as the contact area. The large gear is set as the Contact and the small gear as the Target. The augmented Lagrangian method is chosen as the contact algorithm. Considering that the contact area has friction, the friction coefficient is selected as 0.2 under normal working conditions.
By using the finite – element model simulation results, the distribution values of the tooth – surface stress during the meshing process can be obtained, and the contact – impression area can be determined. According to the actual working conditions, when spiral bevel gears are operating, generally two or more pairs of teeth are meshing simultaneously. For the convenience of observing the change of the stress area, the simulation results of one pair of tooth surfaces under two different loads of 30 N·m and 500 N·m are taken for illustration. When the driving gear rotates clockwise, the stress distributions of the working concave – surface instantaneous contact areas at the starting, middle, and exiting states are obtained, and the tooth – surface stress nephograms are shown in Figure 10.
[Insert Figure 10: Tooth – surface stress nephograms]
4.1.4 Solution Setting
The parameter setting in the transient – dynamics contact analysis is very important. The parameter setting of the load step determines whether the non – linear solution can proceed smoothly. Through analysis and testing, the total time of the load step is set to 0.0125 s, the initial load sub – step is 50, the minimum load sub – step is 20, and the maximum load sub – step is 3000. The large – deformation option is turned on, and an iterative algorithm is used to perform the transient solution for the non – linear contact.
4.2 Simulation Results Analysis and Experimental Comparison
From the simulation results, it can be seen that with the rotation of the spiral bevel gear pair, the tooth – surface contact area moves from the small end to the large end of the tooth. When the load is applied, the tooth surface undergoes elastic deformation. As the load increases, the contact – area size becomes larger, but the overall contact trend does not change due to the change of the load. The contact area is distributed in the middle – small end of the tooth surface and shows a jujube – core shape, forming a certain angle with the tooth – surface direction. Through unit – mesh calculation, the length of the contact area is about 47% of the tooth length, and the height is about 60% of the total tooth height. The simulation results are basically consistent with the design results, indicating that the established model is accurate and reliable.
To further verify the correctness of the model and the simulation results, an actual mating experiment is carried out on the ground spiral bevel gears. A 500 – mm universal rolling inspection machine is used, as shown in Figure 11. Since the universal rolling inspection machine is used to check the overall contact – impression distribution of the contact area, an appropriate load can achieve the experimental purpose. Therefore, the load is set to 30 N·m.
[Insert Figure 11: Universal rolling inspection machine]
The installation position of the spiral bevel gear pair is adjusted by coloring the tooth surface to ensure normal mating. After starting the machine and waiting for the tooth surfaces to mesh normally and run for a certain time, the contact situation of the tooth surface is observed. The contact impression of the large – gear concave surface is obtained, as shown in Figure 12. By analyzing the contact impressions in Figure 10 and Figure 12, it is found that the finite – element simulation results are basically consistent with the experimental results, which verifies the accuracy of the established model and the finite – element contact – analysis results. The contact impressions obtained by simulation and experiment both meet the distribution range of the contact area in the design criteria, indicating that the designed and manufactured spiral bevel gears can meet the actual use requirements.
5. Conclusion
This article comprehensively introduces the parameter modeling and finite – element contact analysis of spiral bevel gears. By analyzing the processing principle of Gleason – system spiral bevel gears, the tooth – surface equations of large and small gears are established, and the tooth – surface discrete points are solved. Through the tooth – surface point cloud, a three – dimensional model of the spiral bevel gear is established, which improves the modeling accuracy.
The finite – element model is established using the transient – dynamics module in ANSYS Workbench. The contact impressions of the large – gear concave surface during the meshing process of the spiral bevel gear are extracted. The simulation results are accurate and reliable. The contact impressions obtained by simulation and experiment meet the requirements of the contact area in the design criteria, indicating that the designed and manufactured spiral bevel gears can meet the actual use requirements.
This research provides a reliable basis for the design, manufacturing, transmission matching, and optimization analysis of spiral bevel gears. Future research can focus on further improving the accuracy of the model, considering more complex working conditions, and studying the influence of manufacturing errors on the performance of spiral bevel gears.
