Abstract
The accessory drive system is an essential power output component of the aero-engine, and the spiral bevel gear is a critical component within this system, ensuring the normal operation of the engine. Under special oil-depleted conditions such as gearbox failure or lubrication system disruption in aviation accessory drive systems, the meshing contact temperature of spiral bevel gear rises rapidly, leading to rapid gear failure due to significant local temperature increases. During this period, the friction mode shifts to dry friction, where the quality of the tooth surface plays a crucial role in determining the flash temperature caused by dry friction. Grinding, as the final process in gear manufacturing, significantly impacts the quality of the tooth surface. To study the ultimate operating time of aircraft under these special conditions, this paper focuses on the spiral bevel gear of an aircraft accessory transmission system. Based on fractal theory, the grinding wheel surface is simulated, and the trajectory of abrasive grain movement is derived. A three-dimensional solid model of the spiral bevel gear considering the surface morphology after grinding is established. Finite element simulation of the simulated gear model is performed to calculate the tooth surface flash temperature, analyzing the relationship between grinding wheel morphology, gear morphology, and tooth surface flash temperature. This realizes a simulation analysis process from computer-generated fractal grinding wheel surfaces to 3D solid gear models considering surface morphology, and then to finite element simulation for tooth surface temperature calculation.
1. Research Background and Significance
The accessory drive system is indispensable for the power output of aero-engines, with spiral bevel gear serving as critical power transmission components ensuring the normal operation of the entire engine. Their surface quality directly affects the tooth surface temperature, which in turn determines the stability and reliability of the entire machine. Currently, most research focuses on fully lubricated conditions, with limited studies conducted under oil-depleted conditions such as gearbox failure or lubrication system disruption in aviation accessory drive systems. Under these special conditions, the significant local temperature rise due to surface quality issues can lead to rapid gear failure.
To ensure the safe return of pilots and passengers, the aircraft’s limit operating time under these conditions should not be less than 30 minutes, requiring the gears to operate continuously under dry friction for a certain period. However, the actual limit operating time of aircraft is currently far below the design limit, becoming a key technical bottleneck restricting China’s development in this field and severely impacting military training and national defense security.
Domestic manufacturing and processing levels lag behind those of foreign countries, resulting in relatively poor tooth surface quality. Grinding, as the final process in gear manufacturing, plays a decisive role in tooth surface quality. Currently, there is limited research on spiral bevel gear grinding processes, and there is a lack of prediction and simulation studies on gear surface roughness. Due to processing inaccuracies, the gear modification design and backlash parameters may not match the processed tooth surface quality. When the backlash parameters are set, if the tooth surface roughness after grinding is lower than expected, it can lead to increased gear meshing vibration and noise, accelerating gear fatigue damage. Conversely, if the tooth surface roughness is higher than expected, it can result in excessive temperatures during gear meshing, causing thermal expansion of the gears and reducing the backlash, which in severe cases can lead to gear jamming. Therefore, improving the overall level of grinding processes, accurately predicting and simulating the surface roughness of gears after grinding, and calculating gear meshing temperature rises are of significant research importance.
2. Research Status
2.1 Spiral Bevel Gear Grinding Technology and Grinding Wheel Research
Spiral bevel gear is classified into circular arc tooth bevel gears, extended hypocycloid tooth bevel gears, and quasi-involute tooth bevel gears based on their tooth line shapes. Due to their high coincidence factors and ability to change transmission directions, spiral bevel gear is widely used in gear transmission systems in aviation, automotive, and marine fields. As a major manufacturing country, China’s grinding technology not only represents our processing capabilities but also reflects our comprehensive capabilities in defense and aerospace fields.
In 1913, the Gleason Company in the United States developed the first spiral bevel gear machining tool. Subsequently, companies such as Oerlikon in Switzerland and Klingelnberg in Germany followed suit, developing CNC gear grinding machines for spiral bevel gear. In China, Central South University successfully developed the first seven-axis five-linkage full CNC gear grinding machine, YK2045, in 2002. In 2010, Hunan Zhongda Chuangyuan CNC Equipment Co., Ltd. manufactured the world’s largest seven-axis five-linkage full CNC spiral bevel gear grinding machine, which was successfully mass-produced and widely accepted by customers.
2.2 Grinding Wheel Surface Morphology Simulation Research
To study the effect of grinding processes on gear surface morphology, it is necessary to simulate the grinding wheel surface morphology. The most accurate method for generating grinding wheel surfaces for kinematic simulations is to measure the grinding wheel surface using measurement equipment and simulate it based on the extracted data. However, this method is time-consuming and requires specific measurement equipment. Therefore, a numerical simulation method that can simulate the grinding wheel surface morphology is needed to replace the actual measured surface.
Many researchers use simple geometric shapes to simulate abrasive grains on the grinding wheel surface. For example, Mingxia Kang and Lu Zhang simulated abrasive grains as arbitrary pentagons composed of five random vertices, while Haonan Li and Tianbiao Yu proposed a combination of geometric shapes, namely truncated octagons, to describe grain shapes based on statistical results of 500 grains measured from real grinding wheels. Other researchers have simplified abrasive grains on the grinding wheel surface into simple geometric shapes such as spheres, ellipsoids, cones, triangles, tetrahedrons, pyramids, and octahedrons. The main advantage of this method is that the required information, such as the nominal diameter of abrasive grains and the distance between them, can be easily derived from grinding wheel specifications. However, it ignores the fact that the shapes of abrasive grains are often complex.
Additionally, some researchers have applied time series, wavelet transforms, Fourier transforms, and Gaussian distributions to simulate grinding wheel surface morphologies. Still, these methods have limitations. Some scholars have applied fractal theory to simulate grinding wheel surface morphologies. Zhang Y and Luo Y used the box-counting method to calculate the fractal dimension and roughness parameters of different grinding surfaces. However, current research on simulating grinding wheel surface morphologies using fractal theory is still in its early stages, with room for further development.
2.3 Research on Predicting Ground Surface Morphology
Grinding plays a decisive role in the quality of the tooth surface as the final process in gear manufacturing. To improve gear surface quality, many scholars have studied gear grinding processes and simulated the microscopic tooth surface morphology models after grinding, aiming to optimize grinding processes. Salisbury reported a grinding wheel modeling method based on measured grinding wheel surfaces and two-dimensional Fourier transforms. By calculating the interference between wheel points and workpiece surfaces at each moment and continuously updating the height of workpiece surface points, the final workpiece surface is calculated.
Wang Y Z and Chen Y Y considered the discrete generation motion relationship between the tool and workpiece and established a calculation model for the surface roughness of spiral bevel gear. They analyzed and described the tooth surface generation process based on equations related to the maximum surface profile height and surface roughness Ra. Ding H and Tang J proposed an innovative collaborative optimization design considering the geometric topology of the tooth surface and load contact performance evaluation. They developed a high-order topological expression and simulation processing model for NC ground tooth surfaces based on the concept of universal motion (UMC) and machine setting characteristics.
In summary, most studies have not considered the impact of vibrations caused by grinding wheel imbalance on tooth surface morphology during the grinding process.
2.4 Research on Tooth Surface Flash Temperature
Currently, the primary method for calculating tooth surface flash temperature is the flash temperature method. Xue Jianhua combined dynamic loads and transient TEHL models to calculate the temperature distribution in the contact area over time. Zhou C and Xing M proposed a thermal network model to predict the contact temperature of spur gears. By dispersing teeth into multiple temperature elements connected by thermal resistance for heat conduction and convection, they established a thermal network model for gear body temperature and flash temperature.
Shi X and Lu X studied several typical gear pairs and found that rougher gear surfaces lead to higher friction and tooth surface temperatures, while smoother surfaces result in lower friction and temperatures. Gan L and Xiao K proposed a method to calculate tooth surface temperature rises by combining numerical calculations and simulation analysis. They loaded the calculated sliding speeds, friction coefficients, and other parameters onto the gear meshing surface as heat loads and applied time- and position-varying heat flux and convection coefficient boundary conditions to a single-tooth finite element model to calculate the flash and bulk temperatures of gears under various lubrication conditions over time.
In summary, methods for calculating flash temperatures roughly fall into three categories: the flash temperature formula method, the finite element simulation method, and a combination of both. However, current research on calculating flash temperatures under oil-depleted conditions using the finite element method employs smooth gear solid models, which differ from the actual meshing contact between micro-protrusions on the tooth surface. Therefore, there are deficiencies in existing research methods.
3. Main Research Content
Based on the analysis of the current research status, the following deficiencies have been identified: existing studies have determined the fractal dimension and fractal roughness parameters of grinding wheel surfaces but have only the study of fractal dimension and fractal roughness parameters when simulating grinding wheel surfaces based on fractal theory. The simulation of spiral bevel gear grinding surface morphology mainly assumes that abrasive grains are simple geometric shapes. Currently, there is limited consideration of the impact of grinding wheel vibrations on tooth surface morphology. Additionally, research on calculating flash temperatures under oil-depleted conditions using the finite element method employs smooth gear solid models, which differ from actual meshing contacts.
To address these deficiencies, this paper focuses on spiral bevel gear in accessory drive systems. Based on fractal theory, grinding wheel surfaces are simulated, considering the impact of grinding wheel vibrations on tooth surface morphology. Rough spiral bevel gear 3D solid models are established and used to replace traditional smooth models in finite element simulations, enabling precise calculations of tooth surface temperatures. The specific research contents are as follows:
- Grinding Wheel Surface Morphology Modeling: Measure the grinding wheel surface morphology using a 3D non-contact profiler. Calculate the fractal dimension and fractal roughness parameters of the grinding wheel profile using the structure function method based on the measured data. Derive the Weierstras-Mandelbrot fractal function for the 3D surface morphology of the grinding wheel. Simulate different grinding wheel surface morphologies using the control variable method and determine the independent variable functions by comparing the simulated and measured grinding wheel morphologies. Finally, simulate the grinding wheel surface morphology based on fractal theory and analyze the characteristics of different grit size grinding wheel surface morphologies.
- Establishment of Rough Gear 3D Solid Models after Grinding: Establish a tooth surface model based on the grinding trajectory of spiral bevel gear. Discretize the tooth surface and project it onto an axial section. Establish equations for the motion trajectories of grinding grains considering vibrations. Discretize the gear and grinding wheel surfaces and simulate tooth surface morphologies under different grinding parameters by combining the grinding theory and Matlab iteration to select the lowest discrete points. Import the tooth surface data points into Solidworks software to generate 3D solid models.
- Analysis of the Influence of Grinding Parameters on Ground Surfaces: Extract tooth surface morphology data using the coordinate transformation method. Change the default coordinate system of the 3D solid gear model to align it with the coordinate system used for measuring the tooth surface with a 3D non-contact profiler. Process spiral bevel gear using a gear grinding machine and measure the tooth surface morphology using a 3D non-contact profiler to verify the established gear surface morphology. Analyze the influence of various grinding parameters on the tooth surface morphology using orthogonal experimental and extreme difference analysis methods. Establish a regression prediction model for grinding surface roughness using multiple regression analysis with a power function model.
- Calculation of Tooth Surface Flash Temperatures: Calculate parameters such as relative sliding speeds and convection heat transfer coefficients using numerical methods. Combine these parameters with finite element software to calculate the total normal force and thermal loads, which are then applied as external loads to the gear meshing surface. Set heating and cooling load steps to calculate the tooth surface flash temperature. Verify the feasibility of combining numerical calculations and finite element methods to calculate tooth surface temperature rises using finite element simulation methods. Analyze the tooth surface temperature rise law and determine the temperature rise values at any point on the tooth surface.
4. Grinding Wheel Surface Morphology Modeling Based on Fractal Theory
The grinding wheel surface morphology significantly impacts the gear surface morphology. Therefore, accurately simulating the grinding wheel surface morphology is crucial for simulating the workpiece surface morphology. This chapter attempts to establish the grinding wheel surface morphology using fractal geometry. The structure function method is employed to calculate the fractal dimension and fractal roughness parameters of the grinding wheel profile based on measured grinding wheel surface data. Combined with the improved W-M fractal function, different grinding wheel surface morphologies are simulated using the control variable method by varying the independent variable functions. The simulated grinding wheel surface morphologies are compared with the measured ones to determine the independent variable functions f(x,y) and g(x,y), and finally, the grinding wheel surface morphology is simulated based on fractal theory. The measured and simulated grinding wheel surface morphologies are compared to verify the feasibility of the simulation method.
4.1 Overview of Fractal Theory
Fractal theory was introduced by the American mathematician Mandelbrot. It provides a mathematical way to describe complex and random shapes. Koch utilized fractal theory to design the classic Koch curve, where a line segment is divided into three equal parts, and an equilateral triangle is formed with the middle segment as its base. The middle line segment is removed, and each remaining segment is repeatedly divided and processed, resulting in the Koch curve. Over the following decades, researchers have proposed various fractal functions, such as the Sierpinski triangle, the Mandelbrot set, and the Koch snowflake.
4.2 Calculation and Analysis of Fractal Parameters for Grinding Wheel Surface Morphology
The fractal dimension (or fractional dimension) is one of the indicators used to analyze complex systems. Fractal measurements can identify nonlinear relationships and scale-invariant behavior in data. Fractals typically have fractional dimensions, which lie between integer values. Compared to traditional spatial descriptions, the fractal (or fractional) dimension is more like a measure of density or the rate of growth towards infinity.
Fractals can have different dimensions, such as the Hausdorff dimension, compass dimension, box dimension, mass dimension, and area-perimeter dimension. Several methods can be used to calculate these dimensions. For self-similar fractals, these dimensions are relatively easy to calculate. Jiang Jiandong and Chen Jin experimentally observed the micro-morphology of grinding wheel surfaces and concluded that the surface profiles exhibit self-affinity. For self-affine curves like surface profiles, special methods are required to determine their dimensions. This paper uses the structure function method to calculate the fractal dimension Dt and fractal roughness parameter G of the profile.