1. Introduction
Spiral bevel gears are crucial components in mechanical transmissions, widely applied in various complex mechanical equipment such as automobiles, aerospace, and ships due to their advantages like large ，low noise, smooth transmission, high efficiency, and strong load – bearing capacity. The grinding process of spiral bevel gears is a complex, nonlinear, and dynamic procedure. In practical manufacturing, it is difficult to measure key processing parameters such as grinding temperature, grinding force, and abrasive wear of the workpiece. This has hindered in – depth research on the grinding mechanism of spiral bevel gears. Therefore, using the finite – element method to simulate the tooth surface grinding process and analyze the influence of different grinding conditions on tooth surface performance is of great significance for reasonably selecting grinding process parameters and optimizing the manufacturing process of spiral bevel gears.
Previous studies have used different analysis methods to simulate the variation laws of the temperature field and grinding force during the grinding process. However, most of them focused on single – particle grinding analysis, while the actual grinding process involves the interaction of multiple abrasive grains. To further explore the complex mechanism of the spiral bevel gear grinding process, this paper considers the correlation between abrasive grains during grinding, takes the spiral bevel gear as the grinding object, and establishes a compound abrasive grain model. Through finite – element simulation and experimental verification, it reveals the variation laws and mechanisms of various factors in the actual grinding process, providing a theoretical basis for optimizing grinding process parameters.
2. Grinding Principle of Spiral Bevel Gears
The grinding process of spiral bevel gears is similar to the complex meshing process of a generating gear and the gear being processed. It can be regarded as the meshing motion without clearance between the imaginary generating gear and the gear being cut. Grinding is a dynamic and nonlinear multi – element coupling process. With the subtle changes of basic grinding motion parameters and abrasive grain size, physical quantities such as the tooth surface temperature field and grinding force of spiral bevel gears will also change, and the mutual interference between abrasive grains needs to be considered.
The process of a grinding wheel grinding the workpiece surface is equivalent to a high – speed cutting process of a milling cutter with densely arranged cutting teeth. From the perspective of the finite – element method, the grinding process can be simplified as countless fine abrasive grains performing cutting motions on the tooth surface. The material removal process of abrasive grains cutting the tooth surface can be divided into three stages: sliding friction, ploughing action, and chip formation (as shown in Table 1).
| Stage | Description | Phenomenon |
|---|---|---|
| Sliding friction | The grinding wheel abrasive grain initially contacts the tooth surface with a small grinding depth and slides over the tooth surface. | Only elastic deformation occurs on the tooth surface. |
| Ploughing action | As the grinding continues, the grinding depth increases, causing the normal and tangential grinding forces to increase. | The tooth surface material begins to undergo plastic deformation, with bulges appearing in front of and on both sides of the abrasive grain. Grooves are scratched on the tooth surface, and the extrusion and friction between the abrasive grain and the tooth surface are intense, with a significant increase in grinding heat. |
| Chip formation | When the grinding depth continues to increase and the temperature reaches or exceeds the critical temperature of the tooth surface material. | Part of the tooth surface material clearly slides along the shear plane to form chips, which accumulate 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 simplify the grinding wheel abrasive grains into regular geometric shapes. This paper selects a conical shape as the basic shape of the abrasive grain. The height of the abrasive grain \(h = 70\mu m\), and the apex angle \(2\theta=108^{\circ}\). To obtain the mutual interference effect between multiple abrasive grains, for new – type grinding tools, the positions and densities of abrasive grains can be pre – arranged. Two abrasive grains with the same parameters are established and connected at the upper half by bonding to achieve side – by – side cutting motion. The abrasive grain size determines the size of the grinding wheel abrasive grains. In this paper, 60 – mesh abrasive grains with intervals of 150μm and 220μm are selected for simulation.
Using the finite – element method to simulate the process of a grinding wheel grinding a spiral bevel gear, the unit body of the tooth surface is set with a length of 2mm, a width of 1mm, and a thickness of 1mm. The basic geometric models of the grinding workpiece and the abrasive grain are established, as shown in Figure 1. [Insert Figure 1: 3D geometry model of compound abrasive grinding here]
4. Establishment of Finite – Element Model and Simulation Analysis
4.1 Establishment of the Finite – Element Analysis Model
The finite – element simulation software is used to simulate the compound abrasive grain grinding process of the spiral bevel gear tooth surface, so as to obtain the influence of different grinding parameters on the tooth surface temperature field, grinding force, and abrasive grain wear. The Lagrangian finite – element method with the previous configuration as the reference configuration is adopted in the analysis process. The greatest feature of this method is that during the solution calculation process, the mesh is re – divided at any time according to the deformation degree of the unit mesh when the tool contacts the tooth surface, avoiding the distortion of the surface mesh and solving the problem of non – convergence of local large – deformation calculations during the grinding process. The simulation links include mesh division, model material setting, workpiece Johnson – Cook material constitutive model, boundary conditions and contact relationship, and solution setting.
4.1.1 Mesh Division
Mesh division is a very important step in finite – element pre – processing, which directly determines the accuracy of the calculation results. To obtain accurate calculation results and improve calculation efficiency, the model is first initially meshed, and the extreme unit lengths of the geometric objects to be meshed are set to 0.05mm and 0.003mm, with a ratio of 2. After the initial mesh division, in order to further ensure the calculation accuracy of the simulation analysis, the mesh of the tooth surface to be ground is locally refined. The refined mesh sizes are 0.02mm and 0.001mm respectively. The tooth surface and the abrasive grain have a total of 162,339 units and 36,389 nodes. The finite – element mesh model is shown in Figure 2. [Insert Figure 2: Finite element mesh model here]
4.1.2 Setting of Model Materials
According to the actual material of the spiral bevel gear, 25CrMo4, the elastic modulus is set to 202GPa, the shear modulus is 78GPa, the Poisson’s ratio is 0.25, and the density is generally \(7.8g/cm^{3}\). The SG grinding wheel is mainly composed of \(Al_{2}O_{3}\), with advantages such as high hardness, good toughness, and strong sharpness.
4.1.3 Workpiece Johnson – Cook Material Constitutive Model
Considering that during the process of abrasive grains grinding the tooth surface, the workpiece is continuously in a state of high stress, high temperature, and large strain, and undergoes thermo – elastic – plastic deformation. Selecting a constitutive model that can express material characteristics is crucial for the accuracy and reliability of the final simulation results. Since the Johnson – Cook material constitutive model has a good expression effect for such problems, it is selected to describe the change relationship between the workpiece material properties and temperature, stress, etc. The specific mathematical expression is: \(\sigma(\varepsilon_{p},\dot{\varepsilon},T)=(A + B\varepsilon_{p}^{n})(1 + C\ln\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}})[(1-(T^{*})^{m}]\) \(T^{*}\equiv(T – T_{s})/(T_{m}-T_{v})\) where \(\sigma\) is stress; \(A\) is the initial yield stress; \(B\) is the strain – hardening parameter; \(C\) is the strain – rate hardening parameter; \(n\) is the hardening exponent; \(m\) is the thermal softening exponent; \(T\) is the workpiece temperature; \(T_{m}\) is the melting point temperature; \(T_{r}\) is the room temperature; \(\varepsilon_{p}\) is the strain. The Johnson – Cook model plastic parameters of material 25CrMo4 are \(A = 120MPa\), \(B = 891MPa\), \(C = 0.02\), \(n = 0.2\), and \(m = 0.64\). During the compound abrasive grain grinding of the tooth surface, when the stress, strain, etc. of the material exceed the set range, chip separation can occur. The shear failure criterion based on the Johnson – Cook constitutive model can accurately provide a theoretical basis for separation and is applicable to such large – deformation simulation analyses.
4.1.4 Boundary Conditions and Contact Relationship
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 the friction between the abrasive grain and the surface to be ground in the actual grinding process, the shear friction type is selected with a friction coefficient \(\mu = 0.6\). Since friction inevitably involves heat conversion, the upper surface of the gear micro – element and the entire abrasive grain are defined as heat – exchange surfaces. The convective heat – transfer coefficient during the simulation is \(0.02N/(s\cdot mm\cdot^{\circ}C)\), and the initial temperatures of the abrasive grain and the workpiece material are set to room temperature, \(20^{\circ}C\).
4.1.5 Solution Setting
The simulation step size needs to comprehensively consider factors such as the amount of operation, calculation time, and simulation accuracy. If the step size is too small, it will cause calculation redundancy; if it is too large, it will lead to low – accuracy results 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, 0.001mm/step, saved every 5 steps, and the total load step is 2000 steps. The simulation type is Lagrangian Incremental, and the Direct Iteration method is selected. Compared with the Newton – Raphson method, this iteration method can obtain better convergence results. The Usui wear model for continuous processing is used to calculate the compound abrasive grain wear, and the calculation formula is: \(W=\int apVe^{-b/T}dt\) where \(W\) is the wear amount; \(P\) is the interface pressure; \(V\) is the sliding speed; \(T\) is the interface temperature; \(a\) and \(b\) are experimental calibration coefficients.
4.2 Simulation Results Analysis
The quality of the spiral bevel gear tooth surface directly affects the operating efficiency, service life, and accuracy of mechanical equipment. The grinding process, as a key link, must strictly control the grinding parameters. Otherwise, cracks such as long – strip cracks, network cracks, and dot – shaped cracks may occur on the tooth surface during the processing or use period. These cracks can cause drastic changes in the maximum contact stress. In actual processing, it is difficult to detect these parameters. Therefore, in order to explore the variation laws of the tooth surface temperature field and grinding wheel force during the grinding process, the control variable method is used to conduct four compound abrasive grain grinding simulation experiments. The specific parameters are shown in Table 2.
| Serial Number | Grinding Speed (m/s) | Grinding Depth (\(\mu m\)) | Abrasive Grain Spacing (\(\mu m\)) |
|---|---|---|---|
| 1 | 30 | 20 | 150 |
| 2 | 50 | 20 | 150 |
| 3 | 30 | 50 | 150 |
| 4 | 30 | 20 | 220 |
4.2.1 Temperature Field Analysis
The tooth surface temperature increases as the abrasive grain grinding progresses. The temperature fields at stable grinding are extracted, and the distributions of the tooth surface temperature fields under different abrasive grain speeds, grinding depths, and abrasive grain intervals are compared, as shown in Figure 3 – Figure 5. [Insert Figure 3: Temperature field distribution at different grinding speeds here] [Insert Figure 4: Temperature distribution under different grinding depths here] [Insert Figure 5: Temperature field distribution under different abrasive grain spacings here] It can be seen from Figure 3 – Figure 5 that under the same grinding depth condition, when the grinding speeds are 30m/s and 50m/s, the maximum tooth surface temperatures are \(819^{\circ}C\) and \(950^{\circ}C\) respectively; under the same grinding speed condition, when the grinding depths are 20μm and 50μm, the maximum tooth surface temperatures are \(819^{\circ}C\) and \(1050^{\circ}C\) respectively; when the other grinding parameters are the same and only the abrasive grain interval is changed, the change in the temperature field is not obvious. The reasons are as follows:
- As the grinding speed increases, the tooth surface temperature rises. This is because with the increase in speed, the abrasive grains increase the effective grinding amount on the tooth surface per unit time, resulting in increased friction work and more heat accumulation, thus raising the grinding temperature.
- As the grinding depth deepens, the tooth surface temperature increases significantly. This is because as the grinding depth increases, 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, leading to a sharp increase in friction force. Therefore, the grinding heat changes significantly. During the continuous grinding process, the temperature may even approach the melting point of the material. At the same time, it can be found from Figure 4 that when the grinding depth increases, the heat dissipation of the tooth surface is slower, making it more likely to cause surface burns. Therefore, in actual grinding processing, coolants must be used correctly to timely remove the heat generated during grinding.
- Changing the abrasive grain interval hardly affects the factors influencing the friction force, 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, it can be clearly seen from the compound abrasive grain grinding model that under the interference of abrasive grains, the temperature fields radiate each other. For example, in Figure 4(b), the temperature of the tooth surface that has not undergone contact grinding can also reach above \(500^{\circ}C\); as shown in Figure 5, due to the interference of abrasive grains on the temperature field, the heat dissipation rate of the tooth surface is also different. These phenomena are consistent with the grinding heat flux distribution under actual grinding conditions.
4.2.2 Grinding Force Analysis
The grinding force is a main parameter that reflects the basic characteristics and laws of the grinding process. It is closely related to the changes of 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 are conducive to a further understanding of the grinding mechanism and are the basis for improving the grinding process. The data of the grinding force (normal and tangential) during the entire grinding process are extracted for comparative analysis. The change curves of the grinding force (normal and tangential) under different abrasive grain speeds, grinding depths, and abrasive grain intervals are studied, as shown in Figure 6 – Figure 8. [Insert Figure 6: Variation curves of grinding force at different speeds here] [Insert Figure 7: Variation of grinding force under different depths here] [Insert Figure 8: Variation curves of grinding force under different intervals here] It can be seen from Figure 6 – Figure 8:
- As the abrasive grains gradually enter the tooth surface material, the grinding force increases to the maximum value with a large gradient. The normal grinding force is always greater than the tangential grinding force, and the change trends of the two are almost the same.
- During the stable grinding stage, the grinding force fluctuates up and down around the maximum value and stabilizes within a fixed range. As the abrasive grains leave the material, the grinding force slowly decreases to zero.
- The average values of the grinding force with the change of speed are shown in Table 3. As the speed increases, the tooth surface temperature rises, and the grinding force decreases. Comparing Figure 7(a) and Figure 7(b), it can be seen that when the grinding depth increases, the grinding force changes significantly. The above simulation results are basically consistent with the experimental results in the literature. | Speed (m/s) | Mean of Normal Force (N) | Mean of Tangential Force (N) | |—|—|—| | 30 | 17.4040 | 9.8559 | | 50 | 16.8783 | 9.3735 |
- The results corresponding to individual time steps in the figure fluctuate severely. This may be due to bad points generated by the distortion of the mesh re – division during the simulation process. It has little impact on the change trend of the grinding force and can be ignored.
Combined with the actual situation, the large negative rake angle of the abrasive grains leads to the normal grinding force being greater than the tangential grinding force. At the same time, the conical shape makes the workpiece material flow to both sides, reducing the resistance of the abrasive grains in the forward direction. This is the essential reason why the grinding depth significantly affects the grinding force. Different from the single – abrasive – grain model, to further explore the interference effect of abrasive grains on the grinding force, the variances of the grinding force data when the abrasive grain intervals are 150μm and 220μm corresponding to Figure 8 are calculated, as shown in Table 4. By comparing the variances of the normal force and the tangential force, it can be seen that the smaller the distance between the abrasive grains, the stronger the interference effect of the grinding force, resulting in poor stability of the grinding process. This
| Abrasive Spacing (\(\mu\)m) | Variance of Normal Force | Variance of Tangential Force | Stability |
|---|---|---|---|
| 150 | 9.2639 | 2.8441 | Worse |
| 220 | 7.7975 | 2.2797 | Better |
4.2.3 Abrasive Grain Wear Analysis
The grinding wheel experiences high – temperature, high – speed, and high – stress conditions when grinding the workpiece. Subtle changes in grinding parameters can easily affect the grinding quality, leading to frequent tool dressing. This 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 for optimizing the actual grinding process. To visually analyze the degree of abrasive grain wear, the wear nephogram after the simulation grinding is extracted, as shown in Figure 9. [Insert Figure 9: Abrasive wear nephogram here]
It can be seen from Figure 9 that regardless of how the grinding parameters change, the main wear surface of the abrasive grain is always along the movement direction of the contact area with the workpiece material. In addition, due to the conical shape, the wear surface diffuses in an elliptical shape, and the diffusion rate gradually accelerates as the grinding continues. Comparing Figure 9(a) and Figure 9(b), it can be obtained that as the grinding speed increases from 30m/s to 50m/s, the wear amount increases from 1.72μm to 2.14μm, and the overall wear area is almost the same. Comparing Figure 9(a) and Figure 9(c), it can be seen that as the grinding depth increases from 20μm to 50μm, the wear amount increases from 1.72μm to 2.5μm, and the wear area expands outward. Comparing Figure 9(a) and Figure 9(d), it is found that the abrasive grain interval has little effect on the change of the wear amount. Therefore, for the grinding process, the grinding depth has the greatest impact on the wear of the grinding tool, followed by the grinding speed, and the abrasive grain interval has the least impact.
5. Tooth Surface Grinding Experiment
To further verify the accuracy of the established model and the finite – element simulation results, the control variable method is used. Four groups of grinding workpieces and grinding wheels with basically the same material, geometric dimensions, processing accuracy, and other parameters are prepared. The grinding speed, grinding depth, and grinding wheel granularity of the grinding wheel are controlled respectively, as shown in Table 5. Each group of spiral bevel gears is subjected to actual grinding experiments under the same working conditions. The first group is used as the control group, and the second, third, and fourth groups are used as the experimental groups for comparative analysis to observe the grinding effect.
| Serial Number | Grinding Speed (m/s) | Grinding Depth (\(\mu\)m) | Grinding Wheel Granularity |
|---|---|---|---|
| 1 | 30 | 20 | Coarse – grained |
| 2 | 50 | 20 | Coarse – grained |
| 3 | 30 | 50 | Coarse – grained |
| 4 | 30 | 20 | Fine – grained |
A Zeiss EVO electron microscope is used to detect and image the tooth surface of the spiral bevel gear after grinding. Figure 10 shows the influence of four different grinding conditions on the ground tooth surface. [Insert Figure 10: Influence of different grinding conditions on tooth surface here]
Comparing Figure 10(a) and Figure 10(b), it is found that as the grinding speed increases, obvious grinding wheel marks appear on the tooth surface. Comparing Figure 10(a) and Figure 10(c), it is found that as the grinding depth increases, the grinding marks deepen further, and at the same time, some sunken damaged surfaces appear on the tooth surface. Comparing Figure 10(a) and Figure 10(d), it is found that as the abrasive grain interval decreases, the grinding marks on the tooth surface are wrinkled, and obvious interference phenomena occur.
The above experimental results are basically consistent with the conclusions obtained from the finite – element simulation, reflecting the accuracy of the established grinding model and the finite – element analysis results. Through simulation and experiments, it is proved that the grinding speed, grinding depth, and abrasive grain interval all have different degrees of influence on the grinding surface performance of spiral bevel gears. Grinding parameters should be strictly controlled during the actual processing process.
6. Conclusion
This paper conducts a simulation analysis of the compound abrasive grain grinding process and verifies it through experiments. It explores the influence of changes in grinding parameters such as grinding speed and grinding depth on the tooth surface temperature field, grinding force, and abrasive grain wear, and draws the following conclusions:
- As the grinding speed and grinding depth increase, the tooth surface temperature will increase significantly, and may even approach the melting point of the material. Especially due to the interference of compound abrasive grains, the tooth surface temperature fields radiate each other, resulting in a high – temperature effect on the non – contact grinding surface and slowing down the heat dissipation rate.
- The smaller the distance between abrasive grains, the stronger the interference effect of the grinding force, resulting in poor stability of the grinding process and making it prone to vibration.
- The grinding depth has the greatest impact on the wear of the grinding tool, followed by the grinding speed. As the speed and depth increase, the wear amount will increase.
- Through comprehensive analysis, it can be seen that during the grinding process, the grinding depth is the most important factor affecting the workpiece temperature, grinding force, and grinding tool wear. It should be strictly controlled during the actual grinding process.
In future research, more complex grinding conditions and material properties can be considered to further improve the accuracy of the simulation model. At the same time, more in – depth experimental research can be carried out to provide more reliable theoretical and practical bases for the grinding process of spiral bevel gears. This research also has certain reference significance for the grinding processing of other similar gears, which can help improve the processing quality and efficiency of the entire gear manufacturing industry.
