Abstract
This paper presents a comprehensive simulation analysis of the cutting force and temperature in gear hobbing process of cycloid gears. By constructing geometric models of the cycloid gear and its corresponding hob, and establishing a thermo-mechanical coupling finite element model, the effects of different hobbing parameters, such as cutting speed and axial feed, were analyzed in detail. The results obtained provide valuable insights into the optimization of gear hobbing process and the improvement of gear accuracy.
1. Introduction
Cycloid gears are widely used in precision reducers such as the RV reducer, where the tooth profile accuracy of the cycloid gear directly affects the transmission accuracy of the reducer. The gear hobbing process is one of the primary methods for processing cycloid gears. However, the cutting force and temperature during gear hobbing process can significantly impact the tooth profile accuracy of the cycloid gear. Therefore, it is crucial to study the influence of gear hobbing parameters on the cutting force and temperature.
2. Modeling of the Cycloid Gear and Hob
2.1 Geometric Modeling of the Cycloid Gear
The cycloid gear tooth profile used in this study is a circular arc-short amplitude epicycloid equidistant tooth profile. The actual meshing process between the cycloid gear and the pin wheel involves enveloping motion, where the base circle of the cycloid gear rolls purely within the base circle of the pin wheel, and the center of the cycloid gear revolves around the center of the pin wheel, while the center of the pin wheel remains stationary.
For simplicity, it is assumed that the cycloid gear is stationary, and the base circle of the pin wheel rolls purely on the base circle of the cycloid gear, with the center of the pin wheel revolving around the center of the cycloid gear. The tooth profile equation of the pin wheel can be derived based on geometric relationships and trigonometric functions,.
2.2 Geometric Modeling of the Hob
Based on the principle of gear meshing and conjugate curve theory, the tooth profile of the cycloid gear and the normal tooth profile of the hob have the same motion relationship as that of a gear and a rack. Therefore, when the pitch circle of the cycloid gear rolls purely on the pitch line of the rack, the tooth profile enveloped by the cycloid gear is the tooth profile of the hob. The hob geometric parameters, such as outer diameter, number of chip pockets, number of hob heads, length, helix angle, and relief amount, can be derived based on the tooth profile equation of the cycloid gear and the geometric relationship between the cycloid gear and the hob, as shown in Table 1.
[Table 1: Hob geometric parameters]
| Hob Parameter | Value |
|---|---|
| Outer Diameter (D) | 75 mm |
| Number of Chip Pockets | 12 |
| Number of Hob Heads | 3 |
| Hob Length (L) | 110 mm |
| Helix Angle (λ) | 3.136° |
| Relief Amount (h) | 5.4 mm |
3. Simulation Process of Cycloid Gear Hobbing
3.1 Material Properties of the Tool and Workpiece
The material of the cycloid gear is 25CrMo4, with a Poisson’s ratio of 0.3 and a density of 7.85×103. The material properties of 25CrMo4, such as Young’s modulus, thermal expansion coefficient, thermal conductivity, and specific heat capacity, vary with temperature, as shown in Table 2. The material of the hob is M35, with a Poisson’s ratio of 0.23, a thermal conductivity of 30 W/(m·K), a specific heat capacity of 15 J/(kg·K), and a density of 8.14×103.
[Table 2: Material properties of 25CrMo4]
| Temperature (°C) | Young’s Modulus (MPa) | Thermal Expansion Coefficient (×10^-5/K) | Thermal Conductivity [W/(m·K)] | Specific Heat Capacity [J/(kg·K)] |
|---|---|---|---|---|
| 20 | 212000 | 1.19 | 41.7 | 6.6189 |
| 100 | 207000 | 1.25 | 43.4 | 3.8936 |
| … | … | … | … | … |
| 1500 | 69440 | 1.49 | 34.1 | 6.1073 |
3.2 Material Constitutive Model and Fracture Criterion
During gear hobbing process, the material undergoes nonlinear deformation and large strains. The Johnson-Cook constitutive model is used in this study to characterize the high strain rate, large strain values, and material softening due to plastic dissipation during gear hobbing process. The constitutive model relationship is given by Equation (11).
In addition, the Johnson-Cook fracture strain model is used to analyze and set the material fracture criterion, considering the intermittent cutting characteristics of gear hobbing process. The expression of the Johnson-Cook fracture strain model is given by Equation (12).
3.3 Finite Element Simulation Model
The geometric models of the tool and workpiece were built in the Deform-3D simulation software. To improve simulation efficiency, the workpiece was cropped, and only the cutting influence zone and adjacent features were retained for cutting simulation. The mesh size of the cutting zone was set to 0.01 mm, and the mesh size of other zones was set to 0.1 mm. The tool was considered rigid due to its much greater stiffness compared to the workpiece and the short analysis time, and motion constraints were applied to the tool according to Equation (9). Axial constraints were applied to the bottom of the workpiece, and symmetric constraints were applied to the cropped surface of the workpiece.
4. Simulation Results and Analysis
4.1 Simulation Results
The simulation was conducted to study the influence of different gear hobbing cutting speeds and axial feeds on the cutting force and cutting temperature during gear hobbing process of the cycloid gear. The gear hobbing parameters used in the simulation are shown in Table 3.
[Table 3: Hobbing parameters]
| Gear Hobbing Parameter | Value Range |
|---|---|
| Cutting Speed (r/min) | 300, 450, 600, 750, 900 |
| Axial Feed (mm/r) | 0.25, 0.5, 0.75, 1 |
The variation curves of the cutting force and cutting temperature with processing time when the cutting speed is 600 r/min and the axial feed is 0.5 mm/r.
4.2 Analysis of the Influence of Processing Parameters
The cutting force curves during gear hobbing process of cycloid gears were derived using the established finite element simulation model, as illustrated in a hypothetical figure (similar to the description provided, though not explicitly in the provided PDF). The analysis reveals distinct trends in the axial and radial cutting forces as they relate to the processing time and parameters.
During the cutting-in stage, the axial cutting force undergoes a rapid increase. This surge is attributed to the initial engagement of the hob with the workpiece material, resulting in a significant increase in the shear area and, consequently, the cutting force. As the cutting process progresses into the stable cutting stage, the axial cutting force fluctuates within a certain range. This stability is maintained due to the consistent engagement of the hob teeth with the workpiece, leading to a relatively constant shear area and cutting force. However, towards the cutting-out stage, the axial cutting force decreases rapidly as the hob teeth disengage from the workpiece.
The radial cutting force exhibits a similar but distinct trend. During the cutting-in stage, it also rapidly increases to a maximum value. However, unlike the axial cutting force, the radial cutting force gradually decreases as the cutting process continues. This decrease can be attributed to the changing geometry of the engagement between the hob and the workpiece, which results in a decreasing shear area and, hence, a decreasing radial cutting force.
The influence of processing parameters, specifically gear hobbing speed and axial feed rate, on the cutting forces was further analyzed. The results indicate that the axial cutting force is relatively less affected by changes in gear hobbing speed. For instance, as gear hobbing speed increases from 300 r/min to 900 r/min, the axial cutting force increases gradually by a small margin. In contrast, the axial feed rate has a more significant impact on the axial cutting force. When the axial feed rate increases from 0.25 mm/r to 0.75 mm/r, the axial cutting force rises slowly. However, a more rapid increase is observed when the axial feed rate increases from 0.75 mm/r to 1 mm/r. Similarly, the radial cutting force also exhibits a notable increase with an increase in the axial feed rate.
These findings suggest that while increasing gear hobbing speed may help in improving production efficiency, it has a limited impact on the cutting forces. On the other hand, increasing the axial feed rate can significantly affect the cutting forces, and therefore, it should be carefully considered to avoid excessive forces that may damage the workpiece or the cutting tool.
In summary, the cutting forces during gear hobbing process of cycloid gears are influenced by both gear hobbing speed and the axial feed rate. Understanding these relationships is crucial for optimizing the cutting parameters to achieve efficient and precise machining of cycloid gears.