This article delves into the crucial aspects of spiral bevel gears in the context of aviation accessory drive systems. It focuses on the significance of grinding processes in determining gear surface quality and the subsequent impact on tooth surface flash temperature under oil – depleted conditions. Through in – depth research and analysis, methods for simulating grinding wheel surface topography, establishing gear tooth surface models, and calculating tooth surface flash temperature are presented, aiming to improve the understanding and performance of spiral bevel gears.
1. Introduction
1.1 Background and Significance
Spiral bevel gears play a vital role in the accessory drive systems of aeroengines. They are key components that ensure the normal operation of the engine. In the event of special oil – depleted conditions such as a broken gear box or a damaged lubrication system in the aviation accessory drive system, the temperature at the meshing contact of spiral bevel gears rises rapidly. This significant local temperature increase can lead to rapid gear failure.
The surface quality of spiral bevel gears directly affects the tooth surface temperature, which in turn determines the stability and reliability of the entire engine. Currently, most research focuses on well – lubricated conditions, while there is a lack of research on oil – depleted conditions. Improving the grinding process of spiral bevel gears, accurately predicting gear surface roughness, and calculating gear meshing temperature rise are of great significance for enhancing the performance and safety of aircraft.
1.2 Research Objectives
The main objectives of this research are as follows:
- To simulate the surface topography of grinding wheels based on fractal theory and establish a three – dimensional surface topography model.
- To establish a tooth surface model of spiral bevel gears considering the influence of grinding wheel vibration during the grinding process and create a three – dimensional solid gear model.
- To analyze the influence of various grinding parameters on the tooth surface quality and optimize the grinding process parameters.
- To calculate the tooth surface flash temperature considering the tooth surface morphology and verify the accuracy of the calculation method.
2. Spiral Bevel Gears: An Overview
2.1 Classification and Applications
Spiral bevel gears can be classified into different types according to their tooth line shapes, such as circular arc tooth bevel gears, extended epicycloid tooth bevel gears, and quasi – involute tooth bevel gears. Due to their large 重合系数 (contact ratio) and ability to change the transmission direction, they are widely used in gear transmission systems in various fields, including aviation, automotive, and marine industries.
| Classification | Characteristics | Applications |
|---|---|---|
| Circular arc tooth bevel gears | Have curved tooth lines in the form of circular arcs | In high – load – bearing and high – speed transmission scenarios in automotive differentials |
| Extended epicycloid tooth bevel gears | Tooth lines are based on extended epicycloid curves | In some precision – transmission mechanical devices |
| Quasi – involute tooth bevel gears | Resemble involute gears in tooth profile characteristics | In certain industrial machinery with specific meshing requirements |
2.2 Manufacturing Challenges
The manufacturing of spiral bevel gears faces several challenges. In China, although there have been certain achievements in the development of spiral bevel gear processing machine tools, there is still a gap compared with foreign countries. The quality of the tooth surface is affected by factors such as manufacturing accuracy and grinding process parameters. If the tooth surface roughness does not meet the requirements, it can lead to problems such as increased gear meshing vibration, noise, and temperature, shortening the service life of the gear.
3. Grinding Wheel Surface Topography Modeling
3.1 Fractal Theory Basics
Fractal theory, proposed by Mandelbrot, can describe complex and random shapes in a mathematical way. Classic fractal curves, such as the Koch curve, demonstrate the self – similarity and complexity of fractal structures. In the context of grinding wheel surface topography, fractal theory can be used to analyze and model the irregularities on the grinding wheel surface.
| Fractal Curve | Generation Process | Characteristics |
|---|---|---|
| Koch Curve | Divide a line segment into three equal parts, and replace the middle part with an equilateral triangle’s side. Repeat this process for each segment. | Self – similar at different scales, with a non – integer dimension |
3.2 Calculation of Fractal Parameters
The structure function method is used to calculate the fractal parameters of the grinding wheel surface. The fractal dimension \(D_f\) and fractal roughness parameter G can be obtained by analyzing the relationship between the structure function \(s(\tau)\) and the wheel profile height function \(Z(x)\). The formula \(S(\tau)=\left<[z(x+\tau)-z(x)]^{2}\right>=C \tau^{\left(4 – 2D_{z}\right)}\) is used, and through logarithmic transformation and curve fitting, the values of \(D_f\) and G can be determined.
3.3 Modeling the Grinding Wheel Surface
Based on the calculated fractal parameters, a three – dimensional rough surface topography model of the grinding wheel is constructed using the improved Weierstras – Mandelbrot (W – M) fractal function. By adjusting the independent variable functions \(f(x, y)\) and \(g(x, y)\) and comparing the simulated surface topography with the measured one, the optimal model is obtained. The simulation results show that as the grinding wheel grain size increases, the fractal dimension D increases, the fractal roughness parameter G decreases, and the grinding wheel surface becomes more complex and fine.
| Grain Size | Fractal Dimension D | Fractal Roughness Parameter G | Surface Morphology Features |
|---|---|---|---|
| SG46 M/# | 1.2747 | 4.3842e – 5 | Relatively large – grained, with less complex surface topography |
| SG180 M/# | 1.3503 | 2.4262e – 5 | Fine – grained, more complex and smoother surface |
4. Spiral Bevel Gear Tooth Surface Modeling
4.1 Tooth Surface Model Establishment Based on Grinding Trajectory
The tooth surface model of spiral bevel gears is established based on the grinding processing trajectory. For small and large spiral bevel gears, different coordinate systems are established, and the tooth surface equations and normal equations are derived using the principle of generating method processing. The grinding process involves the relative movement of the grinding wheel and the gear, and the meshing equation is used to describe the contact relationship between them.
| Gear Type | Coordinate System Setup | Tooth Surface Equation |
|---|---|---|
| Small spiral bevel gear | Multiple coordinate systems including machine tool coordinate system \(S_{m1}\), 摇台坐标系 \(S_{c1}\), etc. | \(r_{l}(\theta_{p}, \varphi_{p})\) (derived through a series of coordinate transformations) |
| Large spiral bevel gear | Similar multiple – coordinate – system setup with different parameter values | \(r_{2}(\theta_{g}, \varphi_{g})\) (derived based on specific coordinate relationships) |
4.2 Considering the Influence of Grinding Wheel Vibration
During the grinding process, the vibration of the grinding wheel has an impact on the tooth surface morphology. By establishing the motion trajectory equation of a single abrasive grain considering vibration, the influence of vibration on the tooth surface can be analyzed. The motion trajectory equation of the i – th abrasive grain \(F_{i}\) in the global coordinate system oxyz is \(z_{G_{i}}=\frac{r_{G_{i}}\left(x_{G_{i}}-\frac{30 v_{w}}{\pi n_{s}}(i – 1)\phi\right)^{2}}{2\left(r_{G_{i}}+\frac{30 v_{w’}}{\pi n_{s}}\right)^{2}}+(r_{l}-r_{i})+A\sin\left(2\pi\frac{f}{v_{w}}x+\varphi\right)\), where A is the vibration amplitude, f is the vibration frequency, and \(\varphi\) is the initial phase.
4.3 Creating the Solid Gear Model
The solid gear model is created by discretizing the tooth surface and the grinding wheel surface. The discrete points on the tooth surface are calculated using Matlab, and then imported into Solidworks to generate a three – dimensional solid gear model. This model takes into account the tooth surface roughness and the influence of grinding parameters, providing a basis for further analysis of gear performance.
5. Experimental Analysis of Spiral Bevel Gear Grinding Surface Topography
5.1 Experimental Setup
The experiment uses a Gleason Phoenix 600G CNC gear grinding machine to process spiral bevel gears. The grinding wheel used is a straight – cup SG grinding wheel, and the gear material is 20CrMnTi with a hardness of 58 – 62HRC. The basic parameters of the spiral bevel gears and the grinding process parameters are shown in the following tables.
| Gear Type | 齿数 | 模数 | 齿宽 | 压力角 | 螺旋角 | 轴线角 | 节锥角 | 节圆直径 | 齿顶高 | 齿根高 | 旋向 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Large gear | 44 | – | 18mm | 20° | 35° | 90° | 71°18′ | 111.76mm | 1.28mm | 3.51mm | 左 |
| Small gear | – | – | – | – | – | – | 18°82′ | 38.10mm | 3.03mm | 1.76mm | 右 |
| Grinding Parameter | Small Gear | Large Gear |
|---|---|---|
| Knife Disk Nominal Diameter | 120mm | 120mm |
| Tool Angle | 22° | 22° |
| Initial Machine Tool Angle | 298°32′ | 77°53′ |
| Radial Setting | 51.67mm | 51.09mm |
| Blank Offset | 1.64mm | – 2.16mm |
| Machine Tool Root Angle | 17°54′ | 17°54′ |
5.2 Measuring Tooth Surface Topography
A three – dimensional non – contact optical profilometer, Taylor Map CCI, is used to measure the tooth surface topography. To measure the tooth surface roughness accurately, the gear is cut into single teeth, and the measurement area is selected as the central area of the tooth width and height. A method for measuring the tooth surface topography of a three – dimensional solid gear model is proposed, which involves changing the coordinate system of the gear model to be consistent with the measurement coordinate system of the profilometer.
5.3 Experimental Results and Analysis
The simulated tooth surface topography is compared with the measured one. The results show that the error between the simulated and measured tooth surface roughness is within 5%, verifying the feasibility of the proposed method. Through orthogonal experiments and range analysis, the influence degree of each grinding parameter on the tooth surface quality is determined. The results show that the grinding depth has the greatest influence on the surface roughness, followed by the grinding wheel grain size, grinding wheel speed, and 展成速度. The optimal grinding process parameters are obtained as a grinding depth of 0.02mm, a grinding wheel speed of 52.5m/s, a grinding wheel grain size of 80M/#, and a 展成速度 of 4.4m/min.
| Grinding Depth (mm) | Grinding Wheel Speed (m/s) | Grinding Wheel Grain Size (M/#) | 展成速度 (m/min) | Experimental Tooth Surface Roughness \(R_{a}\) (μm) | Simulated Tooth Surface Roughness \(R_{a}\) (μm) | Relative Error (%) |
|---|---|---|---|---|---|---|
| 0.02 | 16.5 | 46 | 2.7 | 0.431 | 0.411 | – 4.64 |
| 0.05 | 26.5 | 120 | 7.2 | 0.469 | 0.449 | – 4.26 |
| 0.08 | 35.2 | 46 | 3.6 | 0.621 | 0.594 | – 4.35 |
| 0.11 | 52.2 | 60 | 4.4 | 0.739 | 0.713 | – 3.52 |
6. Tooth Surface Flash Temperature Calculation and Analysis
6.1 Calculation Method
The tooth surface flash temperature calculation combines the numerical calculation method and the finite element analysis method. In oil – depleted conditions, the flash temperature is mainly generated by the friction heat of the micro – convex bodies on the tooth surface. The heat generation formula \(Q = f\times F_{N}\times v_{t}\) is used, and the normal force \(F_{N}\) is calculated through finite – element contact analysis. The sliding speed \(v_{t}\) is calculated based on the kinematics of the contact points on the gear teeth.
6.2 Heat Transfer Parameter Setting
The heat transfer in the spiral bevel gear mainly occurs through convection heat transfer with the air in oil – depleted conditions. The tooth surface is divided into different types, such as meshing tooth surfaces, inner and outer end faces of the teeth, and tooth tips, tooth roots, and non – meshing surfaces. Different convection heat transfer coefficient calculation formulas are used for different types of surfaces.
| Tooth Surface Type | Convection Heat Transfer Coefficient Formula |
|---|---|
| Meshing tooth surface | \(H_{m}=0.228\lambda pr^{1 / 3}d^{-0.269}\left(\frac{V}{v}\right)^{0.731}\) |
| Inner and outer end faces of the teeth | \(H_{n}=0.308\lambda(m + 2)^{0.5}pr_{n}^{0.5}D_{n}^{-0.5}\left(\frac{V_{n}}{v}\right)^{0.5}\) |
| Tooth tips, tooth roots, and non – meshing surfaces | \(H_{r}\) is between 30% – 50% of \(H_{n}\) |
6.3 Simulation and Results
ANSYS software is used to simulate the friction heat generation of the rough gear model. The simulation results show that the highest temperature on the large gear tooth surface is mainly concentrated in the area between the pitch line and the tooth root, while the temperature on the small gear tooth surface is mainly concentrated in the area between the pitch line and the tooth tip. The minimum temperature area on the entire meshing trajectory is near the pitch line. The calculated results of the theoretical and simulation – combined method and the finite – element simulation method are in good agreement, verifying the feasibility of the calculation method.
| Calculation Method | Maximum Temperature Location on Large Gear | Maximum Temperature Location on Small Gear | Maximum Node Temperature (°C) | Maximum Elliptical Contact Temperature (°C) |
|---|---|---|---|---|
| Theoretical and simulation – combined method | Between pitch line and tooth root | Between pitch line and tooth tip | 121 | 106 |
| Finite – element simulation method | Between pitch line and tooth root | Between pitch line and tooth tip | 118 | 106 |
7. Conclusion
7.1 Summary of Research Achievements
- A three – dimensional surface topography model of the grinding wheel based on fractal theory is successfully established, and the relationship between grinding wheel grain size and fractal parameters is obtained.
- A tooth surface model of spiral bevel gears considering the influence of grinding wheel vibration is established, and a three – dimensional solid gear model is created.
- A method for measuring the tooth surface topography of a three – dimensional solid gear model is proposed, and the grinding process parameters are optimized through experiments.
- A calculation method for tooth surface flash temperature considering tooth surface morphology is proposed, and the accuracy of the method is verified through simulation.
7.2 Future Research Directions
Although this research has achieved certain results, there are still areas for improvement. In the future, research can be carried out to consider factors such as surface plastic bulge, chip build – up, and elastic deformation during the grinding process to establish a more accurate three – dimensional solid gear model. Additionally, further studies on the influence of different lubrication conditions on tooth surface flash temperature can be conducted to provide more comprehensive theoretical support for the design and manufacturing of spiral bevel gears.
