1. Introduction
1.1 Research Background and Significance
Spiral bevel gears play a crucial role in various mechanical systems, especially in the accessory drive systems of aeroengines. They are essential for power transmission and ensuring the normal operation of engines. In an aeroengine, the accessory drive system is responsible for powering various components such as fuel pumps, oil pumps, and generators. Spiral bevel gears, as key components in this system, directly affect the performance and reliability of the entire engine.
Under normal operating conditions, the lubrication system in the gearbox ensures smooth gear meshing and reduces wear and tear. However, in special oil – depleted conditions, such as when the gearbox is breached or the lubrication system is damaged, the situation changes dramatically. In these scenarios, the friction mode of spiral bevel gears turns into dry friction. The temperature at the meshing contact area of the gears rises rapidly, which can lead to rapid gear failure. This not only affects the normal operation of the engine but also poses a significant threat to flight safety. For example, in military aircraft operations, sudden gear failures due to oil – depleted conditions can cause mission failures or even endanger the lives of pilots.
Moreover, the surface quality of spiral bevel gears is of utmost importance. The surface quality is mainly determined by the grinding process, which is the final and decisive step in gear manufacturing. A high – quality surface can reduce friction and wear, improve the efficiency of power transmission, and extend the service life of the gears. On the contrary, a poor – quality surface can lead to increased friction, higher temperatures, and accelerated gear failure.
In addition, accurate prediction of the ultimate operating time of aircraft under oil – depleted conditions is a critical issue. Currently, the actual limit operating time of aircraft is far lower than the designed value, which has become a “bottleneck” restricting military training and national defense security. Therefore, in – depth research on spiral bevel gears, including surface morphology modeling, understanding the influence of surface quality on gear performance, and calculating tooth surface flash temperature, is of great significance for improving the reliability and safety of aircraft, as well as enhancing the overall combat effectiveness of the military.
1.2 Research Objectives
The main objective of this research is to establish a comprehensive understanding of spiral bevel gear grinding tooth surface micro – morphology and accurately calculate the tooth surface flash temperature under oil – depleted conditions. Specifically, it aims to:
- Develop a method for modeling the surface morphology of grinding wheels based on fractal theory. This involves simulating the grinding wheel surface, calculating relevant parameters such as fractal dimension and roughness, and validating the accuracy of the simulation method.
- Establish a three – dimensional solid model of spiral bevel gears considering surface morphology after grinding. This requires taking into account the influence of various factors such as grinding process parameters and vibration on the tooth surface morphology, and using appropriate software to generate accurate models.
- Propose a method for measuring the tooth surface morphology of three – dimensional solid gear models. This method should be able to accurately measure and analyze the tooth surface, and provide a basis for comparing simulated and real – world tooth surface morphologies.
- Analyze the relationship between grinding wheel morphology, gear morphology, and tooth surface flash temperature. By using numerical calculation and finite – element simulation methods, calculate the tooth surface flash temperature during gear meshing, and explore the influencing factors and laws of temperature changes.
- Optimize the grinding process parameters of spiral bevel gears. Through experimental analysis and data processing, determine the optimal combination of parameters such as grinding depth, wheel speed, wheel grain size, and generating speed to improve the surface quality of gears.
1.3 Research Significance
- Theoretical Significance
- This research enriches the theoretical system of spiral bevel gear manufacturing. By applying fractal theory to the simulation of grinding wheel surfaces and considering the influence of various factors on tooth surface morphology, it provides a new perspective and method for studying gear manufacturing processes.
- The proposed methods for calculating tooth surface flash temperature and establishing regression models for surface roughness contribute to the development of gear tribology and manufacturing process optimization theory.
- Practical Significance
- The results of this research can be directly applied to the design and manufacturing of spiral bevel gears in the aerospace industry. Optimized grinding process parameters can improve the surface quality of gears, reduce the risk of gear failure under oil – depleted conditions, and enhance the reliability and safety of aircraft.
- The accurate calculation of tooth surface flash temperature provides a basis for gear thermal design and heat treatment processes, helping to improve the overall performance and service life of gears.
2. Literature Review
2.1 Spiral Bevel Gear Grinding Technology and Grinding Wheel Research Status
Spiral bevel gears are widely used in various fields due to their unique advantages such as large 重合度 and the ability to change the transmission direction. They are classified into different types according to the shape of the tooth line, including circular – arc tooth bevel gears, extended epicycloid tooth bevel gears, and quasi – involute tooth bevel gears.
In the field of spiral bevel gear grinding technology, foreign companies have made significant progress. The Gleason Company in the United States developed the first spiral bevel gear processing machine in 1913. Swiss Oerlikon Company and German Klingelnberg Company also have advanced design and production technologies and have developed CNC gear grinding machines for spiral bevel gears. In contrast, although China has made some achievements in recent years, such as the successful development of the seven – axis five – link full – CNC gear grinding machine YK2045 by Central South University in 2002 and the world’s largest seven – axis five – link full – CNC spiral bevel gear grinding machine by Hunan Zhongda Chuangyuan CNC Equipment Co., Ltd. in 2010, there is still a certain gap compared with foreign countries.
2.2 Grinding Wheel Surface Morphology Simulation Research Status
Simulating the surface morphology of grinding wheels is crucial for understanding the grinding process and predicting the surface quality of workpieces. Currently, there are several methods for simulating grinding wheel surfaces. One common method is to use simple – shaped abrasive grains to simulate the grinding wheel surface. For example, some researchers simulate abrasive grains as pentagons, octahedrons, or spheres. While this method is simple and convenient, it ignores the complex shape of actual abrasive grains.
Another approach is to use time – series methods, wavelet and Fourier transform methods, or assume that the height of abrasive grains follows a certain distribution (such as Gaussian distribution) to model the grinding wheel surface. In addition, some scholars have applied fractal theory to the simulation of grinding wheel surfaces. Although progress has been made in these studies, there are still some limitations. For example, the function for simulating three – dimensional rough surfaces using fractal theory is not yet mature, and most research only focuses on analyzing fractal dimensions and topological constants.
| Simulation Method | Advantages | Disadvantages |
|---|---|---|
| Using simple – shaped abrasive grains | Easy to obtain information from grinding wheel specifications | Ignores the complex shape of actual abrasive grains |
| Time – series methods | Can analyze the change trend of grinding wheel surface over time | Complex calculation and limited accuracy |
| Wavelet and Fourier transform methods | Can handle complex surface signals | Require high – level mathematical knowledge and computing power |
| Fractal theory – based methods | Can describe complex and irregular surfaces | Immature three – dimensional rough surface simulation function |
2.3 Prediction of Grinding Morphology Research Status
To improve the surface quality of gears, many scholars have studied the gear grinding process and simulated the microscopic tooth surface morphology after grinding. Some methods consider the interference between the grinding wheel and the workpiece surface at each moment to calculate the final workpiece surface. Others take into account factors such as material accumulation and the influence of tool – workpiece motion relationships on surface roughness. However, most of these studies do not consider the influence of vibration during the grinding process on the tooth surface morphology.
| Researcher | Method | Consideration of Vibration |
|---|---|---|
| Salisbury | Based on measured grinding wheel surface and 2D Fourier transform | No |
| Zhou W | Incorporate material accumulation into the prediction model | No |
| Wang Y Z and Chen Y Y | Establish surface roughness calculation model considering tool – workpiece motion | No |
2.4 Tooth Surface Flash Temperature Research Status
Calculating the tooth surface flash temperature is important for evaluating gear performance and predicting gear failure. Currently, the main methods for calculating tooth surface flash temperature include the flash temperature formula method, finite – element simulation method, and the combination of formula and finite – element simulation method. However, the existing research on calculating tooth surface flash temperature under oil – depleted conditions using the finite – element method mostly uses smooth gear solid models, while in actual situations, the meshing contact of tooth surfaces is the contact of micro – convex bodies on the tooth surfaces, which leads to certain inaccuracies in the calculation results.
| Calculation Method | Advantages | Disadvantages |
|---|---|---|
| Flash temperature formula method | Simple and direct calculation | Limited accuracy, difficult to consider complex factors |
| Finite – element simulation method | Can consider complex geometric shapes and boundary conditions | High – requirement for computing power, inaccurate when using smooth models |
| Combination of formula and finite – element simulation method | Complementary advantages of both methods | Complex calculation process, still some inaccuracies |
3. Modeling of Grinding Wheel Surface Morphology Based on Fractal Theory
3.1 Fractal Theory Overview
Fractal theory, proposed by Mandelbrot, is a powerful tool for describing complex and irregular shapes. It can quantitatively characterize the self – similarity and scale – invariance of objects. Classic fractal curves such as the Koch curve, Koch snowflake, Mandelbrot set, and Sierpinski triangle are typical examples of fractal theory applications. The Koch curve is formed by dividing a line segment into three equal parts, removing the middle part, and replacing it with two line segments of the same length to form an equilateral triangle. Repeating this process continuously results in a complex and self – similar curve.
[Insert an image of the Koch curve here to help readers understand the concept of fractal curves]
3.2 Calculation and Analysis of Fractal Parameters of Grinding Wheel Surface Morphology
3.2.1 Calculation of Fractal Parameters Based on the Structure Function Method
For the surface profile of a grinding wheel, which has self – affine properties, the structure function method is used to calculate its fractal dimension \(D_{f}\) and fractal roughness parameter G. The relationship between the structure function \(s(\tau)\) and the grinding wheel profile height function \(Z(x)\) is expressed as \(S(\tau)=\left<[z(x+\tau)-z(x)]^{2}\right>=C \tau^{\left(4 – 2D_{z}\right)}\), where \(\tau\) is an arbitrary increment, \(D_{t}\) is the fractal dimension of the two – dimensional grinding wheel surface profile, and C is a proportionality coefficient. By taking the logarithm of both sides of the equation, a linear relationship between \(\log S(\tau)\) and \(\log(\tau)\) is obtained. The slope of the straight line in the logarithmic coordinate system is used to calculate the two – dimensional fractal dimension D, and the intercept of the line with the y – axis is the fractal roughness parameter G.
3.2.2 Measurement of Grinding Wheel Surface Morphology and Determination of Fractal Parameters
In this study, the Taylor Map CCI three – dimensional non – contact optical profilometer is used to measure the surface of the SG grinding wheel. The measured data of the grinding wheel surface height \(Z(x)\) is substituted into the relevant formulas to calculate the structure function of grinding wheels with different grain sizes. Then, through curve fitting using Matlab, the fractal dimension and fractal roughness of each grain – sized grinding wheel are determined. The results show that as the grain size of the grinding wheel increases, the fractal dimension \(D_{t}\) increases, and the fractal roughness parameter G decreases. This indicates that the grains on the grinding wheel become smaller and more closely spaced, and the surface morphology becomes more complex and refined.
| Grinding Wheel Model | Fractal Dimension \(D_{t}\) | Fractal Roughness G |
|---|---|---|
| SG46 M/# | 1.2747 | 4.3842e – 5 |
| SG60 M/# | 1.2909 | 3.9039e – 5 |
| SG80 M/# | 1.3272 | 3.4631e – 5 |
| SG120 M/# | 1.3503 | 2.9030e – 5 |
| SG180 M/# | 1.3068 | 2.4262e – 5 |
3.3 Construction of a Three – Dimensional Rough Surface Morphology Model of the Grinding Wheel Based on Fractal Theory
In 1985, Ausloos M and Berman D H proposed a three – dimensional fractal function representation model for rough surfaces. After substituting relevant parameters and simplifying, the W – M function for simulating three – dimensional rough surfaces is obtained. The main influencing factors of the fractal surface morphology are the independent variable functions \(f(x, y)\) and \(g(x, y)\). By comparing the simulated and measured grinding wheel surface morphologies, the independent variable functions \(f(x, y)=(x^{2}+y^{2})^{1 / 2}\) and \(g(x, y)=\arctan(x / y)\) are determined. Using Matlab software, the three – dimensional surface morphologies of grinding wheels with different grain sizes are simulated. The results show that as the grain size of the grinding wheel increases, the number of micro – convex bodies per unit area increases, and their protrusion height decreases, making the grinding wheel surface smoother, which is consistent with the actual situation.
[Insert images of simulated grinding wheel surface morphologies with different grain sizes here to visually show the differences]
3.4 Summary of This Chapter
This chapter successfully established a three – dimensional surface morphology model of the grinding wheel based on fractal theory. By measuring the grinding wheel surface, calculating fractal parameters, and determining independent variable functions, the feasibility of simulating the grinding wheel surface morphology is verified. The obtained results provide a basis for subsequent research on the influence of grinding wheel surface morphology on the tooth surface morphology of spiral bevel gears.
4. Simulation of Spiral Bevel Gear Grinding Surface Morphology
4.1 Establishment of Tooth Surface Model Based on Spiral Bevel Gear Grinding Trajectory
Spiral bevel gears are processed based on the “imaginary generating wheel” grinding principle. In the modeling process, the grinding of small and large spiral bevel gears is considered separately.
4.1.1 Tooth Surface Mathematical Model of Grinding Spiral Bevel Gear Pinion
When using the generating method to process the small gear of a spiral bevel gear, multiple coordinate systems are established, including the machine tool coordinate system \(S_{m l}(O_{m l}, X_{m l}, Y_{m l}, Z_{m l})\), the coordinate system \(S_{c l}(O_{c l}, X_{c l}, Y_{c l}, Z_{c l})\) fixed to the swivel table, the coordinate system \(S_{p l}(O_{p l}, X_{p l}, Y_{p l}, Z_{p l})\) fixed to the cutter head mounting plane, the auxiliary coordinate system \(S_{a}(O_{a}, X_{a}, Y_{a}, Z_{a})\), and the coordinate system \(S_{l}(O_{l}, X_{l}, Y_{l}, Z_{l})\) fixed to the small gear. Through coordinate transformation and the establishment of meshing equations, the tooth surface equation \(r_{l}(\theta_{p}, \varphi_{p})\) and normal equation \(n_{l}(\theta_{p}, \varphi_{p})\) of the small gear in the \(S_{I}\) coordinate system are obtained.
4.1.2 Tooth Surface Mathematical Model of Grinding Spiral Bevel Gear Gear
Similar to the small gear, when processing the large gear of a spiral bevel gear, a series of coordinate systems are established. By performing coordinate transformation and solving meshing equations, the tooth surface equation and normal equation of the large gear can be determined. This provides a theoretical basis for accurately describing the tooth surface shape of spiral bevel gears during the grinding process.
4.2 Establishment of Solid Model of Spiral Bevel Gear Grinding Surface Morphology
4.2.1 Calculation of Spiral Bevel Gear Grinding Trajectory
The discrete – point interpolation method and constraint – solving method are used to calculate the grinding trajectory of spiral bevel gears. First, the tooth surface is discretized into a grid, and the grid nodes are projected onto an axial section. Then, through a series of geometric calculations and equation – solving, the spatial coordinate values of the discrete nodes of the tooth surface grinding motion trajectory of small and large gears are obtained. This step is crucial for accurately simulating the grinding process and the resulting tooth surface morphology.
4.2.2 Plane Grinding Motion Trajectory Equation Considering Vibration
In the actual grinding process, the vibration of the grinding wheel has a significant impact on the tooth surface morphology. To consider this factor, an assumption and simplification are made. A single – grain cutting model is established, and the motion trajectory equation of a single abrasive grain considering vibration is derived. The motion of the abrasive grain includes rotation and vibration, while the workpiece moves in a straight line. The resulting motion trajectory equation provides a more accurate description of the abrasive grain’s movement during the grinding process.
4.2.3 Generation Process and Results of Spiral Bevel Gear Grinding Surface Morphology Solid Model
Based on the above – mentioned theories, the tooth surface is projected onto the axial section, and the motion trajectory of the grinding wheel abrasive grains is calculated. The discrete nodes on the processing trajectory form a matrix, and the grinding wheel surface is also discretized. By comparing the positions of abrasive grains and tooth surface nodes, the lowest – point values of each tooth surface node during the grinding process are selected and stored. Finally, using Solidwork software, these discrete – point data are imported to generate a three – dimensional solid model of the rough gear after grinding. This model reflects the actual tooth surface morphology after grinding, taking into account factors such as grinding parameters and vibration.
4.3 Summary of This Chapter
In this chapter, a tooth surface model was established based on the grinding trajectory of spiral bevel gears, and the influence of grinding wheel vibration on the tooth surface morphology was considered. By calculating the grinding trajectory and using appropriate software, a three – dimensional solid model of the rough gear after grinding was successfully established. This model can be used to predict the tooth surface morphology under different grinding parameters, providing a basis for optimizing the grinding process and improving the surface quality of spiral bevel gears.
5. Experimental Analysis of Spiral Bevel Gear Grinding Surface Morphology
5.1 Spiral Bevel Gear Grinding and Tooth Surface Morphology Measurement Experiment
The Gleason Phoenix 600G CNC gear grinding machine was used to process spiral bevel gears in this experiment. The straight – cup SG grinding wheel was selected, and the gear material was 20CrMnTi with a hardness of 58 – 62HRC. After processing, the basic parameters of the spiral bevel gears, such as the number of teeth, modulus, tooth width, and pressure angle, were measured and recorded. The grinding process parameters, including the cutter head nominal diameter, tool angle, and initial machine angle, were also determined. These experimental settings and parameter measurements are the basis for subsequent analysis of the tooth surface morphology.
| Gear Parameter | Value (Large Gear) | Value (Small Gear) |
|---|---|---|
| Number of Teeth | 44 | – |
| Modulus | – | – |
| Tooth Width | 18mm | – |
| Pressure Angle | 20° | – |
| Spiral Angle | 35° | – |
| Axis Angle | 90° | – |
| Pitch Cone Angle | 71°18′ | 18°82′ |
| Pitch Circle Diameter | 111.76mm | 38.10mm |
| Addendum Height | 1.28mm | 3.03mm |
| Dedendum Height | 3.51mm | 1.76mm |
| Hand of Helix | Left | Right |
| Grinding Process Parameter | Value (Small Gear) | Value (Large Gear) |
|---|---|---|
| Cutter Head Nominal Diameter | 120mm | 120mm |
| Tool Angle | 22° | 22° |
| Initial Machine Angle | 298°32′ | 77°53′ |
| Radial Setting | 51.67mm | 51.09mm |
| Blank Offset | 1.64mm | – 2.16mm |
| Machine Root Angle | 17°54′ | 17°54′ |
5.2 Experiments on Measuring and Comparing Simulated and Real Tooth Surface Morphologies of Spiral Bevel Gears
5.2.1 Determination of Rough Tooth Surface Morphology after Simulated Grinding
Contact – type measuring instruments are prone to errors and high wear when measuring rough tooth surfaces. Therefore, a three – dimensional non – contact optical measurement method is used in this study. A method for measuring the tooth surface morphology of a three – dimensional solid gear model is proposed. By changing the default coordinate system of the three – dimensional solid gear model through coordinate transformation, the new coordinate system is made consistent with the coordinate system of the tooth surface measured by the three – dimensional non – contact profilometer. This allows for the extraction of tooth surface discrete – point coordinates and the calculation of two – dimensional and three – dimensional roughness values, providing a basis for comparing simulated and real tooth surface morphologies.
5.2.2 Comparative Analysis of Simulated and Measured Real Tooth Surface Morphologies
The Taylor Map CCI three – dimensional non – contact optical profilometer was used to measure the tooth surface morphology. The gear was cut into single teeth for measurement, and the middle area of the tooth surface was selected as the measurement area. The simulated tooth surface morphology under different grinding parameters was compared with the measured real tooth surface morphology. The results showed that the relative error between the simulated and measured tooth surface roughness was within 5%, verifying the feasibility of the proposed method for simulating and measuring tooth surface morphologies.
| Serial Number | Grinding Depth a (mm) | Wheel Speed \(v_{s}\) (m/s) | Wheel Grain Size \(M/\#\) | Generating Speed \(v_{w}\) (m/min) | Experimental Tooth Surface Roughness \(R_{a}\) (\(\mu m\)) | Simulated Tooth Surface Roughness \(R_{a}\) (\(\mu m\)) | Relative Error (%) |
|---|---|---|---|---|---|---|---|
| 1 | 0.02 | 16.5 | 46 | 2.7 | 0.431 | 0.411 | – 4.64 |
| 2 | 0.02 | 26.5 | 60 | 3.6 | 0.412 | 0.392 | – 4.85 |
| 3 | 0.02 | 35.2 | 80 | 4.4 | 0.304 | 0.294 | – 3.29 |
| 4 | 0.02 | 52.2 | 120 | 7.2 | 0.314 | 0.304 | – 3.18 |
| 5 | 0.05 | 16.5 | 60 | 4.4 | 0.441 | 0.428 | – 2.95 |
| 6 | 0.05 | 26.5 | 120 | 7.2 | 0.469 | 0.449 | – 4.26 |
| 7 | 0.05 | 35.2 | 46 | 2.7 | 0.414 | 0.395 | – 4.59 |
| 8 | 0.05 | 52.2 | 80 | 3.6 | 0.456 | 0.437 | – 4.17 |
| 9 | 0.08 | 16.5 | 80 | 7.2 | 0.644 | 0.626 | – 2.80 |
| 10 | 0.08 | 26.5 | 120 | 4.4 | 0.616 | 0.592 | – 3.90 |
| 11 | 0.08 | 35.2 | 46 | 3.6 | 0.621 | 0.594 | – 4.35 |
| 12 | 0.08 | 52.2 | 60 | 2.7 | 0.616 | 0.601 | – 2.44 |
| 13 | 0.11 | 16.5 | 120 | 3.6 | 0.758 | 0.736 | – 2.90 |
| 14 | 0.11 | 26.5 | 80 | 2.7 | 0.69 | 0.668 | – 3.19 |
| 15 | 0.11 | 35.2 | 60 | 7.2 | 0.789 | 0.764 | – 3.17 |
| 16 | 0.11 | 52.2 | 46 | 4.4 | 0.739 | 0.713 | – 3.52 |
5.3 Grinding Experiment Analysis and Process Parameter Optimization
5.3.1 Orthogonal Experimental Analysis of Spiral Bevel Gear Grinding Surface Morphology
The main factors affecting the surface morphology of spiral bevel gears during grinding are grinding depth, wheel speed, wheel grain size, and generating speed. An orthogonal experiment was carried out with these four factors, each at four different levels, and the surface roughness \(R_{a}\) as the optimization target. Through the analysis of the experimental results using the range analysis method, it was found that the grinding depth has the greatest influence on the grinding surface roughness, which is about 10 times that of other factors. The larger the grinding depth, the rougher the gear surface. The factors affecting the grinding surface roughness in descending order are grinding depth, wheel grain size, wheel speed, and generating speed. The optimized configuration of the spiral bevel gear grinding process parameters is a grinding depth of 0.02mm, a wheel speed of 52.5m/s, a wheel grain size of 80M/#, and a generating speed of 4.4m/min.
| Level | Grinding Depth a (mm) | Wheel Speed \(v_{s}\) (m/s) | Wheel Grain Size \(M/\#\) | Generating Speed \(v_{w}\) (m/min) |
|---|---|---|---|---|
| 1 | 0.02 | 16.5 | 46 | 2.7 |
| 2 | 0.05 | 26.5 | 60 | 3.6 |
| 3 | 0.08 | 35.5 | 80 | 4.4 |
| 4 | 0.11 | 52.5 | 120 | 7.2 |
| Serial Number | Factor A (Grinding Depth) | Factor B (Wheel Speed) | Factor C (Wheel Grain Size) | Factor D (Generating Speed) | Optimal Target \(R_{a}\) (\(\mu m\)) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 0.431 |
| 2 | 1 | 2 | 2 | 2 | 0.412 |
| 3 | 1 | 3 | 3 | 3 | 0.304 |
| 4 | 1 | 4 | 4 | 4 | 0.314 |
| 5 | 2 | 1 | 2 | 3 | 0.441 |
| 6 | 2 | 2 | 1 | 4 | 0.469 |
| 7 | 2 | 3 | 4 | 1 | 0.414 |
| 8 | 2 | 4 | 3 | 2 | 0.456 |
| 9 | 3 | 1 | 3 | 4 | 0.644 |
| 10 | 3 | 2 | 4 | 3 | 0.616 |
| 11 | 3 | 3 | 1 | 2 | 0.621 |
| 12 | 3 | 4 | 2 | 1 | 0.616 |
| 13 | 4 | 1 | 4 | 2 | 0.758 |
| 14 | 4 | 2 | 3 | 1 | 0.69 |
| 15 | 4 | 3 | 2 | 4 | 0.789 |
| 16 | 4 | 4 | 1 | 3 | 0.739 |
| Level | A | B | C | D |
|---|---|---|---|---|
| \(K_{1}\) | 0.3653 | 0.5685 | 0.5513 | 0.5378 |
| \(K_{2}\) | 0.4450 | 0.5467 | 0.5645 | 0.5617 |
| \(K_{3}\) | 0.6242 | 0.5320 | 0.5235 | 0.5250 |
| \(K_{4}\) | 0.7440 | 0.5313 | 0.5393 | 0.5540 |
| Range R | 0.3787 | 0.0373 | 0.0410 | 0.0367 |
| Factor Importance | 1 | 3 | 2 | 4 |
| Optimal Scheme | \(A_{1}\) (0.02) | \(B_{4}\) (52.5) | \(C_{3}\) (80) | \(D_{3}\) (4.4) |
5.3.2 Regression Prediction Equation of Spiral Bevel Gear Grinding Tooth Surface Roughness
Since the grinding surface roughness \(R_{a}\) has a non – linear relationship with grinding depth, wheel grain size, wheel speed, and generating speed, a power – function model is used for regression analysis. Through logarithmic transformation of variables, a linear model is obtained for calculating the multiple linear regression equation. The regression prediction equation of the grinding surface roughness \(R_{a}\) of spiral bevel gears is \(R_{a}=1.360021\cdot a^{4.385}\cdot V_{s}^{-0.001119}\cdot M^{-0.000479}\cdot V_{w}^{0.00565}\), which can be used to predict the surface roughness under different grinding parameters.
5.4 Summary of This Chapter
This chapter conducted a series of experiments on the grinding surface morphology of spiral bevel gears. By measuring the tooth surface morphology, comparing simulated and real tooth surface morphologies, and analyzing experimental data, the accuracy of the simulated surface morphology model was verified. The influence of grinding parameters on the tooth surface quality was determined, and the optimal configuration of grinding process parameters was obtained. The regression prediction equation of the grinding surface roughness was also established, providing a reference for improving the surface quality of spiral bevel gears in actual production.
6. Calculation and Analysis of Tooth Surface Flash Temperature Considering Tooth Surface Morphology
6.1 Calculation of Tooth Surface Flash Temperature by Combining Theory and Simulation
6.1.1 Meshing Temperature Calculation Formula Considering Tooth Surface Morphology
In oil – depleted conditions, the tooth surface flash temperature of spiral bevel gears is mainly generated by the friction heat of micro – convex bodies on the tooth surfaces. The total heat generated \(Q = f\times F_{N}\times v_{t}\), where \(v_{t}\) is the relative sliding speed between the meshing surfaces, f is the average friction coefficient, and \(F_{N}\) is the normal force. To calculate the normal force of each micro – convex body accurately, a non – linear finite – element loading contact analysis is carried out using Abaqus software to obtain the normal pressure \(P_{i}\) and the total number of nodes N at the meshing moment of the two – tooth surfaces. The total normal force \(F_{N}=\sum_{i = 1}^{N}(P_{i}\times A_{i})\), and the non – uniform instantaneous heat density \(q_{i}=\beta\times f\times p_{i}\times\left|v^{s}-v^{s}\times n\times n\right|\). Considering the heat distribution during gear meshing and the cyclic process of tooth engagement, the average heat load \(\bar{q}_{i}=\frac{n_{I}\Delta t}{60}q_{i}\) is used to calculate the tooth surface temperature.
6.1.2 Setting of Heat Transfer Parameters
In oil – depleted conditions, the heat dissipation of spiral bevel gears mainly occurs through convection heat transfer between the gear and the air. The tooth surface can be divided into three categories: meshing tooth surfaces, inner and outer end faces of the tooth, and tooth top, tooth root, and non – meshing surfaces. The convective heat transfer coefficients of different surfaces are calculated using different formulas. For example, the convective heat transfer coefficient \(H_{m}\) of the first – type meshing tooth surface is \(H_{m}=0.228\lambda p r^{1 / 3}d^{-0.269}\left(\frac{V}{v}\right)^{0.731}\), where \(\lambda\) is the thermal conductivity of air, \(p_{r}\) is the Prandtl number, V is the pitch – line speed, and d is the average tooth height.
| Tooth Surface Category | Convective Heat Transfer Coefficient Formula |
|---|---|
| Meshing Tooth Surfaces (m) | \(H_{m}=0.228\lambda p r^{1 / 3}d^{-0.269}\left(\frac{V}{v}\right)^{0.731}\) |
| Inner and Outer End Faces of the Tooth (n) | \(H_{n}=0.308\lambda(m + 2)^{0.5}p r_{n}^{0.5}D_{n}^{-0.5}\left(\frac{V_{n}}{v}\right)^{0.5}\) |
| Tooth Top, Tooth Root, and Non – Meshing Surfaces (r) | \(H_{r}\) is between 30% – 50% of \(H_{n}\) |