Spiral bevel gears are widely used in the transmission systems of heavy vehicles and military aircraft due to their advantages of high transmission ratio, low noise, and large transmission torque. Grinding is the final step in gear processing, and the surface topography of the ground gear has a significant impact on its service life and reliability.
Introduction
The surface topography of the ground gear is an important indicator of surface integrity, which plays a crucial role in the wear resistance, fatigue strength, and contact stress of the gear. Therefore, it is essential to study the surface topography of spiral bevel gears to ensure their performance and reliability.
Analysis of Abrasive Particle Kinematics in Spiral Bevel Gear Grinding
To simplify the analysis of the complex material removal process in spiral bevel gear grinding, the following assumptions are made:
- The wheel enveloping surface is trimmed to an ideal shape.
- The influence of wheel vibration on the generation of the workpiece surface topography is not considered.
- When the wheel feeds, the workpiece material in contact with the abrasive cutting edge is completely removed.
- The phenomena of side flow and chip are not considered.
In the grinding process, due to the uneven height distribution of the abrasive particles along the radial direction of the wheel, the cutting trajectories of different abrasive particles are distributed on concentric circles of different diameters. The cutting traces left by several abrasive particles on the gear surface superimpose to form the final grinding surface topography of the tooth surface.
In ordinary plane grinding, the geometric model of the single abrasive particle motion is shown in Figure 1. The wheel coordinate system o’x’y’z’ and the workpiece coordinate system oxyz are established. In Figure 1, oxyz is fixed on the processing surface of the workpiece and coincides with the position of the abrasive particle at the lowest point around the wheel. The motion trajectory aob of the abrasive particle G is synthesized by the rotation of the wheel and the translation of the workpiece. According to the geometric position relationship between the abrasive particle and the workpiece in Figure 1, the trajectory equation of the abrasive particle G is as follows:
where , are the instantaneous coordinates of the abrasive particle G, is the relative angle rotated by the abrasive particle, i is the number of discrete points of the workpiece along the x direction, is the distance from the cutting edge vertex of the abrasive particle to the center of the wheel, is the wheel speed, and is the workpiece speed. The grinding mode is reverse grinding. If the grinding mode is forward grinding, the feed speed direction of the workbench is reversed.
According to the kinematic theory of the grinding process, the motion trajectory of the abrasive particle G is:
To study the influence of adjacent abrasive particles on the surface topography, the motion trajectories of adjacent abrasive particles are analyzed. Assuming that the local coordinate system of the abrasive particle is , and the coordinate origin coincides with the origin of the global coordinate system oxyz, the local coordinate system of the abrasive particle is . The motion trajectory equation of the abrasive particle in the global coordinate system oxyz is:
The motion trajectory equation of the abrasive particle in the local coordinate system is:
where is the distance from the abrasive particle to the center of the wheel.
The time required for the wheel center to move a distance L is equal to the working time interval between two adjacent abrasive particles, so:
where is the angle between adjacent abrasive particles. Thus, .
The motion trajectory equation of the abrasive particle in the global coordinate system oxyz is:
The motion trajectory equation of the i-th abrasive particle in the global coordinate system oxyz is:
In the case of spiral bevel gear grinding, the gear is processed based on the principle of the “imaginary generating gear”. The straight cup wheel is trimmed to match the convex and concave surfaces of the spiral bevel gear. During grinding, the straight cup wheel rotates at a set grinding speed around the spindle, while the spiral bevel gear workpiece moves relative to the straight cup wheel along a specific trajectory at a set generating speed. The convex and concave surfaces of the gear contact the inner and outer sides of the wheel, respectively, to grind the profiles of the convex and concave surfaces of the spiral bevel gear.
To improve the simulation accuracy, a three-dimensional laser scanning microscope is used to measure the surface of the spiral bevel gear. A trajectory line in the grinding direction is selected, and its spatial coordinates are extracted. A polynomial is used to fit the trajectory line, and the fitting equation is as follows:
The fitted equations of the convex and concave helical lines can be obtained according to Equation (10), which are also the motion trajectory lines of the gear workpiece during spiral bevel gear grinding and the motion trajectory of the straight cup wheel during simulation. By coordinate transformation, the motion trajectory of the abrasive particle when grinding the spiral bevel gear can be obtained. The transformation equation is as follows:
Characteristics of Wheel Surface Topography
The microscopic topography of the wheel surface is a key factor affecting the material removal during spiral bevel gear grinding. The surface quality and processing efficiency of the spiral bevel gear workpiece after grinding mainly depend on the size, shape, and distribution state of the abrasive particles. Figure 4 shows the microscopic topography of the SG wheel surface with particle sizes of 80 and 180 taken by a Keyence laser scanning microscope. It can be seen that the abrasive particles are randomly distributed on the wheel surface, and each abrasive particle has an irregular size and shape, resulting in different contact pressures on the spiral bevel gear surface and forming different cutting depths.
The maximum diameter of the abrasive particle for a wheel with a particle size of M can be expressed as:
The average diameter of the abrasive particle can be expressed as:
According to the normal distribution proposed by DOMAN et al., the height of the abrasive cutting edge obeys the normal distribution (, ). A wheel model is established using the normal distribution. In the process of establishing the wheel model, it is assumed that the distribution state of the abrasive particles is determined by the size of the wheel abrasive particles, and the exposed height of the cutting edge is assumed to follow a normal distribution, that is, the distribution function of is as follows:
where is the mean of the normal distribution, ; is the variance of the normal distribution, ; M is the abrasive particle size.
It is also assumed that the average interval between adjacent abrasive particles on the wheel surface is:
where S is the structural parameter of the wheel, representing the volume content of the abrasive particles in the wheel, and , where is the volume percentage of the abrasive particles in the wheel.
The simulated topography of the wheel is shown in Figure 5. It can be seen that the larger the abrasive particle size, the smaller the abrasive particle size and the denser the distribution.
The straight cup wheel grinds the spiral bevel gear by forming grinding. The outer side of the wheel is tangent to the concave surface of the spiral bevel gear, and the inner side is tangent to the convex surface of the spiral bevel gear. By measuring the spiral bevel gear with a three-dimensional laser scanning microscope, the contact contour lines of the concave and convex surfaces perpendicular to the grinding direction are extracted, and the established wheel surface profile is compensated to obtain a wheel surface topography simulation model that matches the actual forming wheel surface.
Simulation of Grinding Surface Topography of Spiral Bevel Gear
After completing the modeling and simulation of the straight cup wheel surface, to realize the simulation of the grinding surface topography of the spiral bevel gear, the tooth surface needs to be discretized. The tooth surface is represented by a topological matrix gmn, where each element g(m, n) represents the z-direction coordinate value corresponding to each grid point (m, n) in the xy plane of the tooth surface in the global coordinate system oxyz, that is, the residual height value of the ground tooth surface after solving. The wheel surface is also discretized and represented by a topological matrix hij, where h(i, j) represents the height of the abrasive particle at the (i, j) position in the circumferential and axial directions of the wheel. The grinding process of the tooth surface can be regarded as the process of i abrasive particles on j xoz sections successively scratching the surface. The calculation trajectories of the first to the i-th abrasive particles on the first to the j-th sections are calculated and stored. At each point g(m, n) on the workpiece plane gmn, there are several trajectory lines corresponding to the value zi(m, n) at this point. Since grinding is a material removal process, the abrasive particle with the smallest z-coordinate value of the trajectory at a specific coordinate position will determine the state of the tooth surface at this point. Therefore, the final surface topography of the gear workpiece after grinding can be expressed as the set of the lowest values of the trajectory lines remaining on the workpiece surface, that is:
The simulation process of the grinding surface topography of the spiral bevel gear is as follows:
- Set the grinding parameters and wheel geometry parameters, define the grid size of the workpiece, and represent it with the topological matrix gmn. Each point gij (i = 1, 2,…, m; j = 1, 2,…, n) on the workpiece plane represents the z-direction height value corresponding to each grid point on the workpiece surface.
- Corresponding the discrete points of the abrasive particles to the matrix elements. According to the corresponding relationship between (x, y) and the subscript (i, j) of the workpiece height matrix gij, judge the corresponding relationship between the discrete point G of the abrasive particle and the grid point of the workpiece. When the coordinates (x, y) of the discrete point G of the abrasive particle just fall on the grid point (i, j) of the workpiece, it means that this point just corresponds to the matrix element gij. Otherwise, find the grid point (i, j) closest to (x, y) and use the matrix element gij at this grid point to correspond to this point.
- Cutting judgment. Compare the height direction coordinate value z of the discrete point G of the selected abrasive particle in the workpiece coordinate system at the current moment with the stored value of the corresponding matrix element gij. If z is smaller, it means that the abrasive particle has cut into the workpiece, and at this time, update the stored value of gij with z. Otherwise, do nothing.
- Draw the three-dimensional surface topography according to the stored data in the matrix gij. The flow chart of the simulation of the grinding surface topography of the spiral bevel gear is shown in Figure 6.
Experimental Verification
The spiral bevel gear grinding experiment is carried out on the Gleason – 600G CNC gear grinding machine using the single – factor experiment method to explore the influence of grinding speed vs, generating speed w, and grinding depth ap on the tooth surface roughness. The Norton straight cup SG wheel is used, and the experimental device is shown in Figure 7. The grinding process parameters of the spiral bevel gear are shown in Table 1. During the processing, each tooth groove corresponds to a set of parameters. After the grinding is completed, the tooth sample is cut along the direction of the large end of the gear to a depth of about 10 mm, and the tooth sample is cleaned in an ultrasonic cleaner. The white light interferometer (CCILITE – M112) is used to measure the tooth surface topography and roughness.
The results of the grinding experiment show that with the increase of the grinding speed and the generating speed, the tooth surface roughness decreases; with the increase of the grinding depth, the tooth surface roughness increases.
The simulation results using the same parameters as the grinding experiment are shown in Figure 10. By comparing the simulation diagram in Figure 10 with the experimental measurement diagram in Figure 8, it can be seen that the tooth surface morphology simulation diagrams of the concave and convex surfaces of the spiral bevel gear in Figure 10a and 10b are generally consistent with the tooth surface morphology after the experiment in Figure 8a and 8c. Figure 10a and 10d show the tooth surface morphology simulation diagrams after grinding at different grinding depths. With the increase of the grinding depth, the surface scratches deepen, which is consistent with the workpiece surface morphology after the experiment in Figure 8a and 8b. Figure 10a and 10e show the tooth surface morphology simulation diagrams after grinding at different grinding speeds. With the increase of the grinding speed, the surface texture becomes denser, which is consistent with the workpiece surface morphology after the experiment in Figure 8c and 8d.
By comparing the simulation and experimental results, the correctness and effectiveness of the abrasive particle motion model for grinding the spiral bevel gear with the straight cup wheel are verified. At the same time, it shows that the numerical simulation method can be used to analyze the influence of different grinding parameters on the grinding surface topography of the spiral bevel gear. When analyzing the influence of a certain processing parameter on the grinding surface topography of the spiral bevel gear, the flexibility of the numerical simulation can effectively exclude other interference factors and accurately and efficiently give the difference between the two. Therefore, the surface creation model based on the grinding abrasive particle trajectory of the spiral bevel gear can be used to predict the processed surface topography and reduce the number of experiments to achieve a broad analysis effect.
Conclusion
- Based on the normal distribution model and the actual size of the abrasive particles on the wheel surface, the wheel surface topography is simulated. According to the principle of grinding kinematics, the cutting motion model of the abrasive particles in the grinding of spiral bevel gears is established. By selecting the interference traces of the effective abrasive particles and the workpiece, the three – dimensional numerical topography of the workpiece surface is generated.
- The microscopic topography characteristics of the workpiece surface obtained by simulation and experiment are basically the same, which verifies the correctness and effectiveness of the proposed grinding surface generation algorithm. From the comprehensive results of the experiment and simulation, it can be seen that increasing the grinding speed and the generating speed, and increasing the abrasive particle size are all conducive to obtaining a good grinding surface topography.
- The model can generate the grinding surface topography of spiral bevel gears with different process parameters and can be used to analyze the generation process of the grinding surface topography of spiral bevel gears and the influence mechanism of each processing parameter on it.
To further analyze the influence of different grinding parameters on the surface topography of spiral bevel gears, the following aspects are considered:
- Grinding Speed vs: As mentioned earlier, increasing the grinding speed can lead to an increase in the number of effective abrasive particles per unit time. This results in a more frequent cutting action on the workpiece surface, reducing the residual height and improving the surface quality. However, it is important to note that very high grinding speeds may cause excessive heat generation, which can have negative effects such as thermal damage to the workpiece and accelerated tool wear. Therefore, there is an optimal range for the grinding speed that needs to be determined based on the specific grinding conditions and requirements.
- Generating Speed w: The generating speed affects the cutting length and residual height of each abrasive particle on the tooth surface. When the generating speed increases, each abrasive particle has a longer cutting length on the tooth surface, which leads to a decrease in the residual height left by adjacent abrasive particles. This results in an improvement in the grinding quality of the tooth surface. However, if the generating speed is too high, it may cause instability in the grinding process and affect the accuracy of the gear profile.
- Grinding Depth ap: An increase in the grinding depth leads to an increase in the undeformed cutting thickness of a single abrasive particle. This causes the abrasive particle to leave deeper scratches on the tooth surface, reducing the grinding surface quality. Therefore, in order to obtain a better surface topography, it is necessary to control the grinding depth within an appropriate range.
- Abrasive Particle Size: Comparing Figure 10a with 10c, it can be seen that when the abrasive particle size increases, the density of the abrasive particles also increases, and the protruding height of the abrasive particles on the wheel surface decreases. This means that during grinding, the number of abrasive particles simultaneously involved in grinding increases, which is beneficial for improving the grinding surface quality. However, using larger abrasive particle sizes may also result in a rougher surface finish, so a balance needs to be struck based on the specific requirements of the gear surface roughness.
Comparison of Simulation and Experimental Results
The comparison between the simulation and experimental results shows that the surface micro-topography features of the workpiece obtained by the two methods are basically consistent. This verifies the correctness and effectiveness of the proposed abrasive particle motion model and the grinding surface generation algorithm. The numerical simulation method provides a valuable tool for analyzing the influence of different grinding parameters on the surface topography of spiral bevel gears. By excluding other interference factors, it can accurately and efficiently reveal the relationship between the processing parameters and the surface topography.
For example, in the simulation, it is possible to observe the changes in the surface topography of the gear under different combinations of grinding parameters. This allows for optimization of the grinding process parameters to achieve the desired surface quality. Additionally, the simulation can help predict potential issues in the grinding process, such as excessive surface roughness or damage, and provide guidance for adjusting the process to avoid these problems.
Applications and Future Research Directions
The research on the grinding surface topography of spiral bevel gears has important applications in the manufacturing industry. By understanding and controlling the surface topography, it is possible to improve the performance, reliability, and lifespan of the gears. This is particularly crucial in applications where the gears are subjected to high loads and demanding operating conditions, such as in heavy vehicles and aerospace systems.
Future research directions could include further refinement of the simulation model to take into account more factors that may affect the grinding process, such as the thermal effects, the wear of the abrasive particles, and the dynamic behavior of the machine tool. Additionally, developing more advanced measurement techniques to accurately characterize the surface topography and validate the simulation results would be beneficial. Furthermore, exploring new grinding technologies and strategies to optimize the surface quality and processing efficiency of spiral bevel gears is an area that warrants further investigation.
In conclusion, the study of the grinding surface topography of spiral bevel gears is a complex but important topic that requires a combination of theoretical analysis, numerical simulation, and experimental verification. By continuing to advance in this area, it is possible to make significant improvements in the quality and performance of spiral bevel gears, contributing to the development of various industries that rely on these critical components.