Spiral bevel gears play a crucial role in mechanical transmission, featuring high，low noise, smooth transmission, high transmission efficiency, and strong load-carrying capacity. To improve processing accuracy, reduce transmission noise, and extend service life, the grinding process is typically adopted to reduce deformation errors. However, the grinding process of spiral bevel gears is a nonlinear and dynamic complex process, making it difficult to measure important processing parameters such as the grinding temperature, grinding force, and abrasive wear of the workpiece in actual processing, which hinders in-depth research on the grinding mechanism of spiral bevel gears. Therefore, using the finite element method to simulate the grinding process is of great significance for exploring the complex grinding mechanism of spiral bevel gears.
This article establishes a finite element simulation model of compound abrasive grain grinding based on the Johnson-Cook force-thermal coupling material constitutive equation, and conducts a comparative analysis of the changes in the tooth surface temperature field, grinding force, and abrasive wear under different grinding speeds, grinding depths, and abrasive grain spacings. The simulation results are verified through experiments. The research results provide theoretical guidance for the study of the grinding surface performance of spiral bevel gears and practical processing.
1. Grinding principle of spiral bevel gears
The grinding process of spiral bevel gears is similar to the complex process of the generating wheel meshing with the machined gear, that is, by imagining the generating wheel and the cut gear to perform a gap-free meshing motion. Grinding is a dynamic and nonlinear multi-factor coupling process. Therefore, with the slight changes in the basic motion parameters of grinding and the abrasive grain size, the physical quantities such as the tooth surface temperature field and grinding force of the spiral bevel gear will also change, and the mutual interference effect between abrasive grains needs to be considered.
The process of abrasive grains cutting the tooth surface to remove materials can be divided into three stages: sliding friction, ploughing, and chip formation. In the sliding friction stage, the abrasive grains of the grinding wheel start to contact the tooth surface, with a small grinding depth, and slide across the tooth surface, causing only elastic deformation on the tooth surface. In the ploughing stage, as grinding continues, the grinding depth increases, resulting in an increase in the grinding normal force and tangential force, and the tooth surface material begins to undergo plastic deformation, with a raised phenomenon appearing in front of and on both sides of the abrasive grains. At this stage, grooves are engraved on the tooth surface, the extrusion and friction between the abrasive grains and the tooth surface are intense, and the grinding heat significantly increases. In the chip formation stage, the grinding depth continues to increase, and the temperature reaches or exceeds the critical temperature of the tooth surface material, and some tooth surface materials obviously slip along the shear plane to form chips, which are accumulated in a vortex shape in front of and on both sides of the abrasive grains.
2. Establishment of the grinding geometric model
To facilitate the study of the complex mechanism of the grinding process, many scholars currently simplify the abrasive grains of the grinding wheel into regular geometric shapes, such as cones, hemispheres, quadrangular pyramids, and prisms. In this paper, a conical shape is selected as the basic shape of the abrasive grains. To obtain the mutual interference effect between multiple abrasive grains, two abrasive grains with the same parameters are established for the new grinding tool that can pre-arrange the position and density of the abrasive grains, and the upper parts of the two abrasive grains are connected together by a bonding method to achieve side-by-side cutting motion. The abrasive grain size determines the size of the abrasive grains of the grinding wheel. In this simulation, 60* abrasive grains with a grain spacing of 150 µm and 220 µm are selected.
The finite element method is used to simulate the process of the grinding wheel grinding the spiral bevel gear. The length, width, and thickness of the tooth surface unit are set as 2 mm, 1 mm, and 1 mm, respectively. The basic geometric models of the grinding workpiece and abrasive grains are established.
3. Establishment and simulation analysis of the finite element model
3.1 Establishment of the finite element analysis model
The Lagrangian finite element method using the previous configuration as the reference configuration is adopted in the analysis process. Its significant feature is that during the solution calculation process, the mesh is redivided at any time according to the degree of deformation of the unit mesh when the tool contacts the tooth surface, avoiding mesh distortion and solving the problem of non-convergence in the calculation of local large deformations during the grinding process. The simulation steps include mesh division, setting the model material, the Johnson-Cook material constitutive model of the workpiece, boundary conditions and contact relationships, and solution settings.
3.1.1 Mesh division
Mesh division is a crucial step in the finite element preprocessing, directly determining the accuracy of the calculation results. Firstly, the model is preliminarily meshed, with the extreme unit length of the geometric object to be divided set as 0.05 mm and 0.003 mm, and the ratio is 2. To further ensure the calculation accuracy of the simulation analysis, after the initial mesh division, the mesh of the tooth surface to be ground is locally refined, with the refined mesh sizes being 0.02 mm and 0.001 mm, respectively. There are a total of 162,339 units and 36,389 nodes in the tooth surface and abrasive grains.
3.1.2 Setting the model material
According to the actual material of the spiral bevel gear, 25CrMo4, the elastic modulus is set as 202 GPa, the shear modulus as 78 GPa, the Poisson’s ratio as 0.25, and the density is generally 7.8 g/cm³. The main component of the SG grinding wheel is Al₂O₃, which has the advantages of high hardness, good toughness, and strong sharpness.
3.1.3 The Johnson-Cook material constitutive model of the workpiece
Considering that the workpiece is in a state of large stress, high temperature, and large strain during the abrasive grain grinding of the tooth surface, and undergoes thermoelastic-plastic deformation, choosing a constitutive model that can express the material properties is crucial for the accuracy and reliability of the final simulation results. The Johnson-Cook material constitutive model has a good expression effect for this type of problem, so it is selected to describe the change relationship between the material properties of the workpiece and temperature, stress, etc. The specific mathematical expression is as follows:
where is the stress; is the initial yield stress; is the strain strengthening parameter; is the strain rate strengthening parameter; is the hardening index; is the thermal softening index; is the workpiece temperature; is the melting point temperature; is the room temperature; and is the strain.
The plastic parameters of the Johnson-Cook model of the material 25CrMo4 are , , , , and . In the process of compound abrasive grain grinding of the tooth surface, the stress and strain of the material exceed the set range, and then the chip separation occurs. The shear failure criterion based on the Johnson-Cook constitutive model can accurately provide the theoretical separation basis, which is suitable for this type of large deformation simulation analysis.
3.1.4 Boundary conditions and contact relationships
According to the material properties, the abrasive grains are selected as rigid bodies, and the tooth surface is defined as a plastic body. The bottom surface of the tooth surface is fixed, and the abrasive grains move along the tangential direction of the tooth surface. Additionally, considering the existence of friction between the abrasive grains and the surface to be ground in the actual grinding process, the shear friction type is selected, with a friction coefficient . Friction inevitably leads to heat conversion, and the upper surface of the gear micro-element and the entire abrasive grain are defined as the heat exchange surface. During the simulation period, the convective heat transfer coefficient is , and the initial temperature of the abrasive grains and the workpiece material is set as the room temperature of 20°C.
3.1.5 Solution settings
The simulation step size needs to comprehensively consider factors such as the calculation amount, calculation time, and simulation accuracy. Too small a step size will cause calculation redundancy, while too large a step size will lead to low result accuracy or even non-convergence of the calculation. After repeated tests, the step size is taken as 1/10 of the minimum mesh size, that is, 0.001 mm/step, and it is saved every 5 steps, with a total load step of 2000 steps. The simulation type is Lagrangian Incremental, and the iteration method is Direct Iteration, which can achieve better convergence results compared to the Newton-Raphson method. The Usui wear model for the continuous processing process is selected to calculate the compound abrasive grain wear, and the calculation formula is as follows:
where is the wear amount; is the interface pressure; is the sliding speed; is the interface temperature; and and are the experimental calibration coefficients, respectively.
3.2 Simulation result analysis
3.2.1 Temperature field analysis
The tooth surface temperature increases with the progress of abrasive grain grinding. Now, the temperature field during stable grinding is extracted, and the distributions of the tooth surface temperature field at different abrasive grain speeds, different grinding depths, and different abrasive grain spacings are compared respectively, as shown in Figures 5 to 7.
It can be seen from Figures 5 to 7 that under the same grinding depth condition, when the grinding speed is 30 m/s and 50 m/s, the highest tooth surface temperatures are 819°C and 950°C, respectively; under the same grinding speed condition, when the grinding depth is 20 µm and 50 µm, the highest tooth surface temperatures are 819°C and 1050°C, respectively; when only the abrasive grain spacing is changed while other grinding parameters remain the same, the change in the temperature field is not very significant. The reasons for this are as follows:
- With the increase in the grinding speed, the tooth surface temperature will increase. This is because with the increase in speed, the abrasive grains will increase the effective grinding amount on the tooth surface per unit time, resulting in an increase in frictional work, more heat accumulation, and an increase in the grinding temperature.
- With the deepening of the grinding depth, the tooth surface temperature will increase significantly. This is because with the increase in the grinding depth, the contact area between the abrasive grain surface and the tooth surface material increases, and the extrusion force on the tooth surface also significantly increases, leading to a sharp increase in friction, so the grinding heat will change significantly, and with the continuous grinding, it even approaches the melting point of the material. At the same time, as can be seen from Figure 6, with the increase in the grinding depth, the tooth surface heat dissipation is slower, which is more likely to cause surface burns. Therefore, coolant must be used correctly in actual grinding to take away the heat generated by grinding in a timely manner.
- Changing the abrasive grain spacing has little effect on the factors affecting friction, so the change in the temperature field is not significant.
- The overall temperature change trend is almost the same as that of the single-particle grinding model, reflecting the correctness of the compound abrasive grain grinding model. However, compared with the single-abrasive grain grinding model, from the compound abrasive grain grinding model, it can be clearly seen that under the interference effect between the abrasive grains, the temperature fields radiate to each other. As shown in Figure 6(b), the temperature of the tooth surface without contact grinding can also reach more than 500°C; as shown in Figure 7, the heat dissipation rate of the tooth surface is also different due to the interference effect of the abrasive grains. These phenomena are consistent with the distribution of the grinding heat flux under the actual grinding conditions.
3.2.2 Grinding force analysis
The grinding force is a main parameter that reflects the basic characteristics and laws of the grinding process, and it is closely related to the changes in the grinding temperature, tooth surface strain, and tool wear. It is an important reason for the consumption of grinding energy, the generation of heat, and the occurrence of grinding vibration. The study and analysis of this parameter are conducive to further understanding the grinding mechanism and lay the foundation for improving the grinding process.
The grinding force (normal and tangential) data in the entire grinding process are extracted for comparative analysis. The change curves of the grinding force (normal and tangential) at different abrasive grain speeds, different grinding depths, and different abrasive grain spacings are studied, and the results are shown in Figures 8 to 10.
It can be seen from Figures 8 to 10 that:
- As the abrasive grains gradually enter the tooth surface material, the grinding force increases to the maximum with a large gradient, and the grinding normal force is always greater than the tangential force, and the change trends of the two are almost the same.
- In the stable grinding stage, the grinding force fluctuates up and down around the maximum value, stabilizes within a fixed range, and gradually decreases to zero as the abrasive grains leave the material.
- The average values of the grinding force with the change in speed are shown in Table 2. With the increase in speed, the tooth surface temperature increases, so the grinding force decreases. By comparing Figures 9(a) and 9(b), it can be seen that as the grinding depth increases, the grinding force will change significantly, and the above simulation results are basically consistent with the experimental results in the literature [19].
- The result fluctuations corresponding to individual time steps in the figure are more severe, which may be due to the bad points caused by the distortion in the mesh redivision during the simulation process. However, it has little influence on the change trend of the grinding force and can be ignored.
By combining the actual analysis, it can be known that the larger negative rake angle of the abrasive grains leads to the grinding normal force being greater than the tangential force, and the conical shape allows the workpiece material to flow to both sides, reducing the resistance of the abrasive grains in the forward direction, which is the essential reason why the grinding depth significantly affects the grinding force.
To further explore the interference effect of the abrasive grains on the grinding force, the variance of the grinding force data corresponding to Figure 10 when the abrasive grain spacings are 150 µm and 220 µm, respectively, is calculated, as shown in Table 3. By comparing the variance of the normal force and the tangential force, it can be seen that the smaller the spacing between the abrasive grains, the stronger the interference effect of the grinding force, resulting in poor stability of the grinding process, which is also an important factor that cannot be ignored in the generation of vibration during the grinding process.
3.2.3 Abrasive grain wear analysis
The grinding wheel grinding the workpiece needs to continuously experience high temperature, high speed, and high stress. Slight changes in the grinding parameters can easily affect the grinding quality, resulting in frequent use of tool dressing, which not only reduces the grinding accuracy of the grinding wheel but also affects the processing efficiency. Therefore, studying the causes and laws of abrasive grain wear is crucial to optimizing the actual grinding process. To visually analyze the degree of abrasive grain wear, the wear nephogram after the end of the simulation grinding is extracted, as shown in Figure 11.
It can be seen from Figure 11 that regardless of the change in the grinding parameters, the main wear surface of the abrasive grains always follows the direction of motion of the contact area with the workpiece material. In addition, due to the conical shape, the wear surface shows an elliptical diffusion, and the diffusion rate gradually increases with the continuous grinding. By comparing Figures 11(a) and 11(b), it can be obtained that with the increase in the grinding speed from 30 m/s to 50 m/s, the wear amount increases from 1.72 µm to 2.14 µm, and the overall wear area is almost the same; by comparing Figures 11(a) and 11(c), it can be seen that with the increase in the grinding depth from 20 µm to 50 µm, the wear amount increases from 1.72 µm to 2.5 µm, and the wear area spreads outward; by comparing Figures 11(a) and 11(d), it is found that the change in the abrasive grain spacing has little effect on the change in the wear amount. Therefore, for grinding processing, the grinding depth has the greatest impact on the tool wear, followed by the grinding speed, and the abrasive grain spacing has the least impact.
4. Tooth surface grinding experiment
To further verify the accuracy of the established model and the finite element simulation results, the control variable method is adopted. Four groups of grinding workpieces and grinding wheels with basically the same parameters such as material, geometric size, and processing accuracy are prepared. The grinding speed, grinding depth, and grinding wheel grit size of the grinding wheel are controlled respectively, as shown in Table 4, and the spiral bevel gears of each group are controlled to perform actual grinding experiments under the same working conditions. Taking the first group as the control group, the second, third, and fourth groups are used as the experimental groups for comparative analysis to observe the grinding effect.
