Spiral bevel gears play a crucial role in mechanical transmission due to their advantages such as large contact ratio, low noise, smooth transmission, high transmission efficiency, and strong load-carrying capacity. However, the grinding process of spiral bevel gears is a nonlinear and dynamic complex process. Exploring the complex grinding mechanism of spiral bevel gears by simulating the grinding process based on the finite element method is of great significance. 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 grain wear under different grinding speeds, grinding depths, and abrasive grain spacings. The simulation results are also verified by experiments.
1. Introduction
Spiral bevel gears (Gleason system) are important parts in mechanical transmission. To improve processing accuracy, reduce transmission noise, and increase service life, the gear grinding process is usually adopted to reduce deformation errors, and its processing route is: milling – heat treatment – gear grinding. As the final process of spiral bevel gear processing, the grinding process is often accompanied by a series of complex plastic deformation processes. In the actual processing process, it is difficult to measure important processing parameters such as the grinding temperature, grinding force, and abrasive wear of the workpiece, which hinders the in-depth study of the grinding mechanism of spiral bevel gears. Therefore, using the finite element method to simulate the grinding process of the tooth surface and analyze the impact of different grinding conditions on the tooth surface performance is of great significance for the rational selection of grinding process parameters and the optimization of the processing and manufacturing process of spiral bevel gears.
Previous simulation studies have adopted different analysis methods to simulate the change rules of the temperature field and grinding force in the grinding process, but most of them are single-particle grinding analyses. The actual grinding process is a process in which multiple abrasive grains interact and repeatedly cut the workpiece, so there must be an interference effect between the abrasive grains. In order to further explore the complex mechanism of the grinding process of spiral bevel gears, this article considers the correlation between the abrasive grains during grinding, takes the spiral bevel gear as the grinding object, establishes a compound abrasive grain model, and uses the finite element method to simulate the grinding process of the SG grinding wheel on the tooth surface. Based on the Johnson-Cook material constitutive model of the workpiece, by comparing and analyzing the overall trends of the temperature field, grinding force, and abrasive grain wear in the grinding area of the workpiece at different grinding speeds, grinding depths, and abrasive grain spacings, the change rules and mechanisms of various factors in the actual grinding process are revealed, providing theoretical guidance for the parameter optimization of the grinding process.
2. 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 processed 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, so 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 the abrasive grains needs to be considered.
The process of the grinding wheel grinding the workpiece surface is a high-speed cutting process with a milling cutter with densely arranged cutter teeth. Using the finite element idea, the grinding process can be simplified as countless fine abrasive grains performing a cutting motion on the tooth surface. Therefore, studying the principle of single abrasive grain grinding is the basis for understanding the grinding processing mechanism.
The material removal process of abrasive grains cutting the tooth surface can be divided into three stages: sliding friction, ploughing, and chip formation. As shown in Figure 1, where is the linear speed of the grinding wheel, is the generating speed, a is the grinding depth, is the tangential grinding force, and is the normal grinding force.
- Sliding friction stage: When the grinding wheel abrasive grain starts to contact the tooth surface, the grinding depth is small, and it slides across the tooth surface, causing only elastic deformation on the tooth surface.
- Ploughing stage: With the continuous grinding, 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. A raised phenomenon appears in front of and on both sides of the abrasive grain, which is the ploughing stage. At this stage, grooves are engraved on the tooth surface, the extrusion and friction between the abrasive grain and the tooth surface are intense, and the grinding heat increases significantly.
- Chip formation stage: The grinding depth continues to increase, 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 grain.
3. Establishment of grinding geometric model
In order 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 article, a cone is selected as the basic shape of the abrasive grain, as shown in Figure 2, where the height of the abrasive grain and the top cone angle . In order to obtain the mutual interference effect between multiple abrasive grains, for new grinding tools that can pre-arrange the position and density of the abrasive grains, two abrasive grains with the same parameters are established, 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 article, the abrasive grain size with intervals of 150 and 220 is selected for simulation.
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, as shown in Figure 3.
4. Establishment and simulation analysis of the finite element model
4.1 Establishment of the finite element analysis model
Using the finite element simulation software to simulate the compound abrasive grain grinding process of the tooth surface of the spiral bevel gear, the influence of different grinding parameters on the tooth surface temperature field, grinding force, and abrasive grain wear can be obtained. The Lagrangian finite element method with the previous configuration as the reference configuration is adopted in the analysis process. The biggest feature of this method is that during the solution calculation process, the mesh is re-divided at any time according to the degree of distortion of the unit mesh when the tool contacts the tooth surface, avoiding the distortion of the surface mesh, which can solve the problem of non-convergence of the calculation of local large deformations in the grinding process. The simulation links include: mesh division, setting the model material, the Johnson-Cook material constitutive model of the workpiece, boundary conditions and contact relationships, and solution settings.
4.1.1 Mesh division
As a very important step in the finite element preprocessing, whether the mesh division is appropriate directly determines the accuracy of the calculation results. In order to obtain accurate calculation results and improve the calculation efficiency at the same time, the model is first divided into meshes initially, and the extreme unit lengths of the geometric objects to be divided are set as 0.05 mm and 0.003 mm, with a ratio of 2. In order to further ensure the calculation accuracy of the simulation analysis, after the initial mesh division is completed, the mesh of the tooth surface to be ground is locally refined, and the refined mesh sizes are 0.02 mm and 0.001 mm, respectively. There are a total of 162,339 units and 36,389 nodes on the tooth surface and abrasive grains, and the finite element mesh model is shown in Figure 4.
4.1.2 Setting the model material
According to the actual material 25CrMo4 of the spiral bevel gear, the elastic modulus is set as 202 GPa, the shear modulus is 78 GPa, the Poisson’s ratio is 0.25, and the density is generally . The main component of the SG grinding wheel is , which has the advantages of high hardness, good toughness, and strong sharpness.
4.1.3 Workpiece Johnson-Cook material constitutive model
Considering that the workpiece is in a state of large stress, high temperature, and large strain during the abrasive grain grinding process of the tooth surface, and thermal elastic-plastic deformation occurs, it is crucial to select a constitutive model that can express the material properties for the accuracy and reliability of the final simulation results. Since the Johnson-Cook material constitutive model has a good expression effect on this type of problem, the Johnson-Cook constitutive is selected to describe the change relationship between the material properties of the workpiece and temperature, stress, etc. The specific mathematical expression is:
where is the stress; is the initial yield stress; is the strain hardening parameter; is the strain rate hardening parameter; is the hardening index; is the thermal softening index; is the workpiece temperature; is the melting point temperature; is the room temperature; is the strain.
The plastic parameters of the Johnson-Cook model of the material 25CrMo4 are ,and . respectively. 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 can occur. 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.
4.1.4 Boundary conditions and contact relationships
According to the material properties, the abrasive grain is selected as a rigid body, and the tooth surface is defined as a plastic body. The bottom surface of the tooth surface is fixed, and the abrasive grain moves along the tangential direction of the tooth surface. In addition, considering that there is friction between the abrasive grain and the surface to be ground in the actual grinding process, the shear friction type is selected, and the friction coefficient . Friction will inevitably have heat conversion, and the upper surface of the gear micro-element and the whole abrasive grain are defined as the heat exchange surface. The convective heat transfer coefficient during the simulation period is , and the initial temperature of the abrasive grain and the workpiece material is set as the room temperature of 20°C.
4.1.5 Solution settings
The simulation step size needs to comprehensively consider factors such as the amount of calculation, calculation time, and simulation accuracy. Too few steps will cause calculation redundancy, and too many steps will lead to low result accuracy or even calculation non-convergence. 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. Compared with the Newton-Raphson method, this iteration method can obtain better convergence results. The Usui wear model of the continuous processing process is selected to calculate the compound abrasive grain wear, and the calculation formula is:
where is the wear amount; is the interface pressure; is the sliding speed; is the interface temperature; and are the experimental calibration coefficients, respectively.
4.2 Simulation results analysis
4.2.1 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 conditions, 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 conditions, when the grinding depth is 20 and 50 respectively, the highest tooth surface temperatures are 819°C and 1050°C respectively; when the other grinding parameters are controlled to be the same, only by changing the abrasive grain spacing, the change of the temperature field is not obvious. The reasons for this are as follows:
- With the increase of the grinding speed, the tooth surface temperature will increase. This is because with the increase of the speed, the abrasive grains will increase the effective grinding amount on the tooth surface per unit time, resulting in an increase in frictional work, causing more heat accumulation, and making the grinding temperature rise.
- With the deepening of the grinding depth, the tooth surface temperature will increase significantly. This is because with the increase of 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 increases significantly, resulting in a sharp increase in friction, so the grinding heat will change significantly, and with the continuous grinding, it will even approach the melting point of the material. At the same time, as can be seen from Figure 6, when the grinding depth increases, the heat dissipation of the tooth surface is slower, which is more likely to cause surface burns. Therefore, in actual grinding processing, coolant must be used correctly to take away the heat generated by grinding in time.
- 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 temperature field is also different due to the interference of the abrasive grains. These phenomena are consistent with the distribution of the grinding heat flow under the actual grinding conditions.
4.2.2 Grinding force analysis
Grinding force is a main parameter that reflects the basic characteristics and rules of the grinding process, and it is closely related to the changes in grinding temperature, tooth surface strain, and tool wear, and is an important cause of grinding energy consumption, heat generation, and grinding vibration. The research and analysis of this parameter is conducive to a further understanding of the grinding mechanism and is the basis for improving the grinding process.
The grinding force (normal and tangential) data in the whole process of grinding is 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:
- The grinding force increases to the maximum value with a large gradient as the abrasive grains gradually enter the tooth surface material, 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, and stabilizes within a fixed range. With the abrasive grains leaving the material, the grinding force slowly decreases until it is zero.
- The average values of the grinding force with the change of speed are shown in Table 2. With the increase of the speed, the tooth surface temperature increases, so the grinding force decreases. By comparing Figure 9(a) and Figure 9(b), it can be seen that when 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 .
- The result fluctuations corresponding to individual time steps in the figure are more serious, which may be due to the bad points caused by the distortion of the mesh re-division in the simulation process, which has little effect on the change trend of the grinding force and can be ignored.
In combination with 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 makes the workpiece material 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. Different from the single abrasive grain model, in order 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 and 220 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 a poorer stability of the grinding process, which is also an important factor that cannot be ignored in the generation of vibration in the grinding process.
