This article focuses on the dynamic change of tooth surface contact of high-precision spiral bevel gears during the meshing process. By analyzing the processing principle of spiral bevel gear tooth surfaces, the tooth surface equation of the spiral bevel gear is established. The tooth surface dispersion point equation is used to accurately calculate the coordinates of each point on the tooth surface, and a three-dimensional model of the spiral bevel gear is constructed. A finite element model of the spiral bevel gear is built, and the transient dynamics module is adopted to analyze the dynamic change of the tooth surface contact stress during the meshing process, obtaining the contact impression on the concave surface of the large spiral bevel gear. Through comparison with the experimental results, it is shown that the proposed method is accurate and reliable, providing a reliable basis for the design, manufacturing, transmission coordination, and optimization analysis of the spiral bevel gear.
1. Introduction
Spiral bevel gears (Gleason system) have significant characteristics such as high transmission efficiency, compact structure, and stable transmission, and are widely used in high-speed and heavy-load equipment like automobiles, construction machinery, aerospace, and ships. The strength and dynamic performance of spiral bevel gears are of great significance in ensuring the smooth operation of mechanical equipment under complex working conditions and reducing vibration and noise. Compared to other tooth-shaped structures, the tooth surface of spiral bevel gears is more complex, making it relatively difficult to establish a real tooth surface finite element model and accurately describe the dynamic change mechanism of the actual spiral bevel gear pair contact. Therefore, establishing an accurate finite element model of the spiral bevel gear and conducting the tooth surface contact dynamic performance analysis under load conditions is very important for the design and processing of the spiral bevel gear.
2. Tooth Surface Modeling of Large and Small Wheels
2.1 Tooth Surface Equation of the Large Wheel
The right-handed large wheel is processed by the generating method, and the tooth surface is a part of the envelope surface of the tool cutting trajectory. By analyzing the processing process, it can be known that the cutter head rotates around the axis to form a cutting cone surface, and the swivel table and the machined large wheel blank also rotate around the axis to form the large wheel tooth surface.
To illustrate the position of the large wheel cutter head, a coordinate system is established as shown in Figure 1. In the figure, is the machine center, is the cutter head center, the plane is contained in the plane where the cutter tip is located (coinciding with the machine plane ), is the axial cutter position, and is the angular cutter position.
To illustrate the position of the large wheel blank, a coordinate system as shown in Figure 2 is established. In the figure, is the designed cone vertex of the wheel blank, is the bed position, is the axial wheel position, and is the wheel blank installation angle.
The coordinate equation of the cutting cone surface is established as follows based on Figure 3:
In the equation, is the coordinate equation of the cutting cone surface, is the cutter tip radius (as shown in Figure 4), and the “±” respectively represent the concave and convex surfaces of processing the large wheel.
The unit normal vector of the large wheel cutting cone surface can be expressed as:
Through the coordinate transformation in equation (3), the cutting cone surface equation can be transformed into the coordinate system fixed to the large wheel blank, and the tooth surface equation and the normal vector of the large wheel can be obtained.
In the equation, is the coordinate transformation matrix, and is the swivel table rotation angle.
By analyzing the large wheel processing process, it can be known that there is no generating motion in the forming method, so is taken as zero, and the tooth surface equation only related to and is obtained.
2.2 Boundary Conditions
The surface determined by the tooth surface equation is a spatial surface with a complex geometric shape. The points required to construct the tooth surface of the spiral bevel gear must be within a specified range, so it is necessary to limit the range of the surface coordinate variables and . Through projection transformation, it can be known that the points on the tooth surface of the large wheel must be located within the planar quadrilateral composed of the tooth tip line, tooth root line, front cone surface, and back cone surface, as shown in Figure 5.
The coordinate of the spatial surface point projected on the plane is calculated as follows:
In the equation, is the axial displacement, is the radial displacement, and is the normal displacement.
Accordingly, it can be judged whether the points calculated by equation (3) meet the boundary conditions, thereby obtaining the range of the surface coordinate variables and . Obviously, this tooth surface equation is a nonlinear equation. The iterative algorithm is used to call the fsolve function in MATLAB to solve the discrete points of the tooth surface, and the algorithm flow is shown in Figure 6.
2.3 Tooth Surface Equation of the Small Wheel
The left-handed small wheel is processed by the cutter tilt method, and the processing principle is basically similar to that of the right-handed large wheel. The positions of the cutter head and the wheel blank are shown in Figures 7 and 8. In the figures, is the machine center, is the cutter head center, is the swivel table angular velocity, and is the swivel table rotation angle. The definitions of other parameters are the same as those of the large wheel.
The tooth surface of the small wheel processed by the cutter tilt method is enveloped by the cutting cone surface of the cutter head, where the swivel table rotation angle is constantly changing. From the meshing equation at the contact point, it can be known that the relative velocity of the cutter head and the wheel blank is perpendicular to the normal vector , and the relationship is as follows:
Finally, the tooth surface equation of the small wheel is obtained, and the boundary conditions and solution methods are similar to the principle of the large wheel tooth surface.
3. Spiral Bevel Gear Modeling
Write the tooth surface equation and other calculation formulas of the right-handed large wheel and left-handed small wheel analyzed above into corresponding programs, and substitute the basic parameters of the spiral bevel gear pair (Table 1) and processing parameters (Table 2) into the program for operation.
Table 1: Basic Parameters of the Gear Pair
| Parameter | Large Wheel | Small Wheel |
|---|---|---|
| Tooth Number | 33 | 28 |
| End Face Module (mm) | 4.8 | 4.8 |
| Outer Cone Distance (mm) | 123.3 | 123.3 |
| Tooth Width (mm) | 37 | 37 |
| Tooth Top Height (mm) | 2.2 | 2.3 |
| Tooth Root Height (mm) | 3.3 | 2.9 |
| Pitch Cone Angle (°) | 49.4 | 40.2 |
| Root Cone Angle (°) | 47.5 | 39.3 |
| Face Cone Angle (°) | 52.5 | 43.1 |
Table 2: Processing Parameters
| Parameter | Small Wheel | Large Wheel | ||
|---|---|---|---|---|
| Concave Surface | Convex Surface | Concave Surface | Convex Surface | |
| Cutter Head Pressure Angle (°) | 20 | 20 | 20 | 20 |
| Cutter Tip Circle Radius (mm) | 1.8 | 2.2 | ||
| Cutter Top Distance (mm) | 2.4 | |||
| Radial Cutter Position (mm) | 93.9 | 93.8 | 92.9 | |
| Angular Cutter Position (°) | 48.4 | 45.5 | 49.3 | |
| Machine Rolling Ratio | 2.2 | 2.3 | 1.3 | |
| Bed Position (mm) | 0.55 | -0.65 | 0 |
- Calculate the discrete points of the convex and concave surfaces of the large wheel according to equation (3), as shown in Figure 9.
- Import the obtained discrete point data into the 3D modeling software SolidWorks in text format, use the obtained point cloud for 3D modeling, and complete the modeling using commands such as curves and surfaces, as shown in Figure 10. Place the discrete points in the same coordinate system to generate the concave and convex surfaces of the large wheel, use surface clipping and shear intersection surfaces to generate the tooth profile of the large wheel, and array the tooth profiles to establish the 3D model of the large spiral bevel gear.
- Similarly, establish the 3D model of the small wheel, assemble the gear pair according to the specified position using the gear mating command in the mechanical mating, and use the interference check function in the SolidWorks software to visually and clearly observe whether the assembly of the spiral bevel gear is reasonable to avoid interference, providing good mating conditions for the next finite element dynamic contact analysis.
4. Finite Element Contact Analysis of the Spiral Bevel Gear
4.1 Establishment of the Finite Element Analysis Model
Based on the ANSYS Workbench platform of the finite element software, the transient dynamics module is used to perform dynamic contact analysis of the spiral bevel gear. The main links include setting the model material parameters, meshing the grid, setting the boundary conditions and contact relationships, and setting the solution parameters. The transient dynamics aims to analyze the dynamic response of the structure under the action of transient or steady loads, and its output result is the stress, strain, and displacement of the contact area changing with time. The nonlinear transient dynamics is used to analyze the contact process of the spiral bevel gear, and the commonly used contact analysis algorithm is the augmented Lagrangian method, which can reduce the contact pressure calculation penetration to an acceptable extent.
- Set the model material parameters.
Import the assembled model into the finite element software Workbench, and set the material parameters according to the actual material 18Cr2Ni4WA of the spiral bevel gear, with the elastic modulus , Poisson’s ratio , and density .
- Mesh the grid.
The accuracy and type of meshing the grid directly determine the accuracy of the 3D model analysis results. An overly dense grid will increase the calculation time and occupy a large amount of computer resources. In order to obtain accurate calculation results and improve calculation efficiency, a tetrahedral mesh model is selected, and the local refinement of the gear contact surface expected to mesh is performed. The mesh size of the refined part is 1 mm, and the mesh size of the non-refined part is 10 mm. The finite element mesh model is shown in Figure 11, with a total of 619,607 nodes and 418,275 elements.
- Set the boundary conditions and contact relationships.
During the operation of the spiral bevel gear pair, the motor applies a torque to the driving wheel through the wheel shaft to rotate around the central axis, and the driving wheel and the driven wheel mesh to drive the driven wheel to rotate around the central axis. The driven wheel not only has an input torque but also a load. Therefore, the applied load is that the driving wheel rotates around the central axis at a speed of 600 r/min. In order to compare the influence of the load on the contact area, the resistance torques of the driven wheel are set as 30 N·m and 500 N·m respectively. Select the locally refined tooth surface as the contact area, set the large wheel as Contact, and the small wheel as Target, and select the augmented Lagrangian method as the contact algorithm. Considering that the contact area is of the type with friction, the friction coefficient is selected as 0.2 under normal working conditions.
Use the finite element model simulation results to obtain the distribution value of the tooth surface stress during the meshing process and obtain the contact impression area. According to the actual working conditions, it can be known that generally, two or more pairs of teeth of the spiral bevel gear mesh simultaneously during operation. In order to facilitate the observation of the stress area change, the simulation results of one pair of tooth surfaces under two different loads of 30 N·m and 500 N·m are taken for explanation. The driving wheel rotates clockwise, and the stress distribution of the instantaneous contact area on the working concave surface is obtained at three state moments of starting, middle, and exiting, and the tooth surface stress nephogram is shown in Figure 12.
- Solution settings.
The parameter settings in the transient dynamics contact analysis are relatively important, and the parameter settings of the load step determine whether the nonlinear solution proceeds smoothly. After analysis and testing, the total load step time is set as 0.0125 s, the initial load sub-step is 50, the minimum load sub-step is 20, the maximum load sub-step is 3000, the large deformation option is turned on, and the iterative algorithm is used for the transient solution of the nonlinear contact.
4.2 Simulation Results Analysis and Experimental Comparison
It can be seen from Figure 12 that as the spiral bevel gear pair rotates, the tooth surface contact area moves from the small end to the large end of the tooth. Due to the elastic deformation of the tooth surface, the contact area area increases with the increase of the load, but the overall contact trend does not change due to the change of the load. The contact area is distributed at the smaller end of the middle of the tooth surface and presents a jujube-shaped, forming a certain angle with the tooth surface direction. Through the calculation of the unit mesh, the contact area length is about 47% of the tooth length, and the height is about 60% of the full tooth height. The simulation results are basically consistent with the design results, indicating that the established model is accurate and reliable.
In order to further verify the correctness of the model and simulation results, a practical mating experiment is carried out on the ground spiral bevel gear. Use a 500 mm universal rolling inspection machine, as shown in Figure 13. Since the universal rolling inspection machine is used to inspect the overall contact impression distribution of the contact area, an appropriate load can achieve the experimental purpose, so the load is set as 30 N·m.
Adjust the installation position of the spiral bevel gear pair through tooth surface coloring to make it mate normally, start the machine, wait for the normal meshing of the wheel tooth surface, operate for a certain period of time, and observe the tooth surface contact situation, and the contact impression on the concave surface of the large wheel is obtained as shown in Figure 14. By analyzing the contact impressions in Figures 12 and 14, it is found that the finite element simulation and the experimental results are basically consistent, verifying the accuracy of the established model and the finite element contact analysis results. The contact impressions obtained through simulation and experiments all meet the distribution range of the contact area in the design criteria, indicating that the designed and manufactured spiral bevel gear can meet the actual use requirements.
5. Conclusion
This article starts from the forming principle of the spiral bevel gear to solve the discrete points of the tooth surface, and establishes a three-dimensional model through the tooth point cloud, improving the modeling accuracy. The finite element model is established by using the transient dynamics module in ANSYS Workbench, and the contact impression on the concave surface of the large spiral bevel gear during the meshing process is extracted. Through analysis, it is found that the simulation results are accurate and reliable.
