Abstract
The cutting force in spiral bevel gear milling is a crucial factor in the design of bevel gear machine tools. It also serves as a fundamental basis for vibration control, tool wear status monitoring, and cutting parameter optimization. This paper investigates the formation method for milling spiral bevel gears, specifically focusing on the large pinion. Based on the principles of gear cutting and geometric theories of spiral bevel gears, the instantaneous chip area is derived. Subsequently, by applying material constitutive equations and oblique angle cutting theory, the shear zone stress is calculated, resulting in the construction of an instantaneous cutting force model. The model’s accuracy is verified through experiments and Matlab simulations.
1. Introduction
The milling of spiral bevel gears is a complex process that requires precision and accuracy to achieve optimal performance. The cutting force during this process significantly impacts the machining quality, tool life, and the overall efficiency of the production line. As such, the development of accurate cutting force models is crucial for optimizing the milling process.
In this study, we delve into the forming method for milling spiral bevel gears, particularly focusing on the large pinion. The objective is to establish a reliable cutting force model that captures the dynamics of the milling process accurately. By understanding the instantaneous chip area and the shear zone stress, we aim to develop a model that can predict cutting forces with high precision.
2. Literature Review
Several approaches have been employed in the literature to model cutting forces in milling processes. These can be broadly classified into three categories:
- Metal Cutting Theory-Based Models: These models express cutting forces as functions of machining parameters such as feed rate, spindle speed, and depth of cut. However, these models often require extensive orthogonal experiments to determine the coefficients associated with the machining parameters [1][2][3].
- Mechanical Models: These discretize the cutting edge of the tool along the axial direction and solve for the cutting forces on each micro-segment. By integrating the forces on these micro-segments, the total cutting force on the tool can be obtained [4].
- Simplified Cutting Force Models: These models treat the cutting force as the product of the cutting area and a cutting force coefficient, where the coefficient depends on the tool and workpiece materials [5][6].
While each of these approaches has its merits, there is a need for a more comprehensive and accurate model that captures the intricacies of the spiral bevel gear milling process.
3. Methodology
This study adopts a multi-step approach to develop and validate an instantaneous cutting force model for spiral bevel gear milling using the forming method. The methodology can be broadly divided into the following stages:
- Derivation of the Instantaneous Chip Area
- Calculation of Shear Zone Stress
- Construction of the Cutting Force Model
- Programming and Simulation
- Experimental Validation
3.1 Derivation of the Instantaneous Chip Area
The instantaneous chip area is a critical component in determining the cutting force during the milling process. In the forming method for spiral bevel gears, the relative positions of the workpiece and the tool are carefully adjusted to ensure precise machining. The chip area varies continuously during the milling process, necessitating a dynamic approach to its calculation.
Table 1: Key Parameters in Chip Area Calculation
| Parameter | Symbol | Description |
|---|---|---|
| Blade generatrix equation | rt | Defines the shape of the tool’s cutting edge |
| Spiral bevel gear face cone | ra | Defines the shape of the workpiece surface |
| Machine tool coordinate system | Mmt | Transformation matrix from tool to machine CS |
| Face cone coordinate system | Mmf | Transformation matrix from face cone to CS |
By establishing the coordinate systems for the tool and the workpiece and transforming the blade and face cone equations into the machine tool coordinate system, the intersection curve can be determined. This curve represents the instantaneous contact between the tool and the workpiece, from which the chip area can be derived.
3.2 Calculation of Shear Zone Stress
The shear zone stress, or the unit cutting force, is another vital parameter in the cutting force model. It is calculated using the Johnson-Cook material constitutive model in conjunction with oblique angle cutting theory. This approach accounts for the effects of strain, strain rate, and temperature on the shear stress in the material.
Table 2: Key Parameters in Shear Zone Stress Calculation
| Parameter | Symbol | Description |
|---|---|---|
| Shear strain | ξ | Deformation of the material in the shear zone |
| Shear strain rate | ξ˙ | Rate of deformation in the shear zone |
| Material constants | A, B, C, m, n | Constants from Johnson-Cook model |
| Temperature effects | Tr, Tm | Reference and melting temperatures |
Using these parameters, the shear stress τ can be calculated as:
τ=31(A+B(3ξ)n)(1+Cln(ξ˙0ξ˙))(1−(Tm−TrT−Tr)m)
3.3 Construction of the Cutting Force Model
The instantaneous cutting force on each tool tooth is determined by the unit cutting force (shear zone stress) and the instantaneous chip area. These forces are decomposed into tangential, radial, and axial components, which are then transformed into the machine tool coordinate system.
Table 3: Decomposition of Cutting Forces
| Force Component | Symbol | Direction |
|---|---|---|
| Tangential | FTqj | Along cutting speed |
| Radial | FNqj | Perpendicular to path |
| Axial | FZqj | Along tool axis |
The total cutting force in each direction is the sum of the forces from all active teeth:
Fx=j=1∑ZH(FTqjsin(θ(t))−FNqjcos(θ(t)))
Fy=j=1∑ZH(FTqjcos(θ(t))+FNqjsin(θ(t)))
Fz=j=1∑ZHFZqj
3.4 Programming and Simulation
A Matlab program was developed to simulate the milling process and calculate the instantaneous cutting forces. The program simulates the movement of the tool and workpiece, determines the active cutting teeth, calculates the chip area and shear zone stress, and finally computes the cutting forces in each direction.
3.5 Experimental Validation
To validate the cutting force model, an experimental setup was configured using a YKH2235 CNC spiral bevel gear milling machine equipped with a Kistler 9171 dynamometer. The experiments were conducted using standard test pieces with varying machining parameters.
Table 4: Experimental Parameters
| Parameter | Value |
|---|---|
| Spindle speed | 120 rpm |
| Feed rate | 20 mm/min |
| Tool diameter | 127 mm |
| Number of teeth ((Z_H)) | 20 |
| Workpiece material | 42CrMo4 steel |
The measured cutting forces were compared with the simulated results to assess the accuracy of the model.
4. Results and Discussion
4.1 Comparison of Experimental and Simulated Cutting Forces
The comparison of experimental and simulated cutting forces in the X, Y, and Z directions is presented in Figures 1-3. The results show good agreement between the two, indicating the validity of the proposed model.
4.2 Error Analysis
The relative errors between the experimental and simulated cutting forces were calculated for each direction. The results are summarized in Table 5.
Table 5: Relative Errors in Cutting Forces
| Direction | Relative Error (%) |
|---|---|
| X | 9.24 |
| Y | 13.18 |
| Z | 13.25 |
The errors are within an acceptable range, demonstrating the robustness and accuracy of the proposed cutting force model.
5. Conclusion
This study presents a comprehensive approach to modeling the instantaneous cutting forces in spiral bevel gear milling using the forming method. By deriving the instantaneous chip area and calculating the shear zone stress, an accurate cutting force model was constructed and validated through experiments. The results indicate that the model predicts cutting forces with high precision, laying a solid foundation for the design of bevel gear machine tools and optimization of milling parameters.
Future work could focus on extending the model to account for tool wear, varying workpiece materials, and more complex cutting conditions. Additionally, real-time monitoring and adaptive control strategies based on this model could further enhance the milling process.