Abstract
Gear shaving is a significant method for gear finishing operations. Compared with gear grinding, it boasts high production efficiency, low processing costs, and the ability to fix radial errors, making it widely adopted in the gear manufacturing industry. However, when using a shaving cutter with a standard involute profile to process gears, a “midconcave” phenomenon appears near the pitch points of the shaved gear tooth profiles. This phenomenon is more pronounced when shaving gears with large moduli and low tooth numbers, significantly impacting the shaving accuracy and gear transmission quality. Therefore, it is essential to delve into the mechanism of the midconcave error in gear shaving and adopt effective measures to resolve it.
1. Introduction
1.1 Background and Research Significance
With the rapid development of related industries, the demand for gears has increased significantly. Simultaneously, the requirements for gear quality and precision have also gradually risen. As an important step in gear finishing, gear shaving demands even higher precision. However, during the shaving process, when using a standard involute helical surface shaving cutter, the tooth profile of the shaved gear exhibits varying degrees of concavity near the pitch circle, known as the midconcave error. The midconcave error is a primary cause of high noise and short service life during gear transmission. Thus, addressing the midconcave issue in gear tooth profiles is crucial for gear production. This research aims to explore the mechanism behind the midconcave error and propose effective improvement methods, drawing on knowledge from gear meshing principles, classical elastic mechanics, material mechanics, and other disciplines, while utilizing engineering software such as Mathematica, UG, and ANSYS for auxiliary research.
1.2 Current Status of Gear Processing Technology
1.3 Current Status of Gear Shaving Technology
1.4 Current Research Status of Gear Shaving Midconcave Error
Gear shaving, as one of the gear finishing methods, boasts high efficiency and low cost but is prone to midconcave phenomena near the meshing nodes during the process, leading to increased noise and shortened gear lifespan. The mechanism behind shaving errors is complex and not yet fully understood, making it a significant challenge in the field of gear shaving. Current research focuses on analyzing the mechanism of the midconcave phenomenon and exploring effective measures to improve it, primarily through two approaches: analyzing the mechanism of the shaving process and seeking effective measures to improve the midconcave error from aspects such as process and tooth profile modification.
Table 1. Summary of Current Research Approaches
| Approach | Description |
|---|---|
| Mechanism Analysis | Analyzing the reasons or influencing factors leading to midconcave |
| Process and Modification Measures | Seeking effective measures to improve the midconcave error |
1.5 Main Research Content
This thesis aims to make substantial progress on the challenge of the midconcave error in gear shaving profiles, focusing on both the formation mechanism and improvement methods. The research combines traditional analytical methods with modern numerical methods for mechanism exploration and conducts in-depth research on negative modified gear shaving, balance shaving, and shaving cutter modification. The main content includes:
- Introducing the basic content of gear shaving, including its principles, processes, cutter classifications, and methods.
- Reviewing the current research status and development of gear tooth profile midconcavity.
- Establishing a mathematical model of the shaving cutter, conducting an analytical study on the spatial meshing of helical gear pairs, deriving meshing equations, and analyzing the influence mechanism of factors such as relative velocity, induced normal curvature, and cutting force on midconcave errors.
2. Exploration of the Formation Mechanism of Gear Shaving Midconcave Error
2.1 Introduction
2.2 Analytical Study of Gear Shaving Meshing
Assuming the tooth surface equations of the shaving cutter and the shaved gear are known, the transformation relationships between the coordinate systems can be derived as follows when the shaving cutter rotates around the z1-axis by an angle φ1 and the shaved gear rotates around the z2-axis by an angle φ2.
The involute helical surface equation of the shaving cutter’s right tooth surface is:
Where rb1 is the base circle radius of the shaving cutter, and θ1 is the involute angle of point M on the end face involute of the shaving cutter’s tooth surface.
To visually represent the position of the contact trace on the shaving cutter tooth surface, specific calculations were conducted using two sets of gear data, as shown in Tables 2.1 and 2.2.
Table 2.1 Parameters of the First Set of Gears
| Parameter | Value |
|---|---|
| … | … |
Table 2.2 Parameters of the Second Set of Gears
| Parameter | Value |
|---|---|
| … | … |
3. Finite Element Contact Analysis of Gear Shaving
3.1 Introduction
3.2 Model Establishment
3.2.2 Parametric Modeling of Helical Gears
To simulate gear performance, such as load capacity and dynamic analysis using computer technology, 3D modeling of the gear is necessary. Although many methods and articles on gear modeling exist, precise methods, especially for the precise modeling of transition curves, are scarce. Although the tooth root transition curve does not contribute to the meshing process, it significantly impacts gear strength, especially bending strength, and must be accurately depicted during analysis.
Using UG’s parameter settings, expression creation and editing, and spreadsheet functions, this thesis achieved complete parametric design of the gear, precisely establishing 3D models of the shaving cutter and the shaved gear.
3.3 Finite Element Analysis
…
4. Negative Modified Gear Shaving and Balance Shaving
4.1 Introduction
4.2 Negative Modified Gear Shaving
4.2.2 Design Calculation
The shaving process is illustrated in Figure 4.1. The shaving cutter rotates clockwise, and a tooth surface of the shaved gear enters meshing from point A at the tooth tip and ends at point C at the tooth root. When the tooth profile of the shaved gear is at point A, the meshing of the shaving cutter’s tooth profile is at point F. When the tooth profile of the shaved gear is at point F, the meshing of the next tooth profile is at point A, and so on. Changes in the modification coefficient inevitably alter the end-face meshing angle. Therefore, when designing the modification coefficient of the shaving cutter, it is determined based on the end-face meshing angle using the no-backlash meshing equation during the shaving process.
<img src=”https://example.com/gear_shaving_meshing_diagram.png” />
Figure 4.1 Diagram of gear shaving meshing
4.3 Balance Shaving
4.3.4 Calculation Example
Using the gear parameters shown in Table 4.1, after calculation, the two modification coefficients of the shaving cutter that satisfy balance shaving are found to be 0.5362 and 0.7635, respectively. After verification, 0.7635 satisfies the restriction condition of Equation (4.17), so it is more reasonable to choose 0.7635 to design the balance shaving cutter. The tooth profiles of the shaved gears obtained using an unmodified conventional shaving cutter and a balance shaving cutter are shown in Figure 4.7 (a) and (b), respectively.
The tooth profile curve of the gear shaved with a conventional shaving cutter, and (b) represents the tooth profile curve of the gear shaved with the designed balance shaving cutter. It can be seen that balance shaving significantly improves the tooth profile accuracy and midconcave error of the shaved gear.
6. Conclusions
This thesis conducted in-depth and detailed research on the formation mechanism and improvement measures of the mid-concave error in gear shaving. By utilizing knowledge of gear meshing principles, elastic mechanics, material mechanics, and finite element analysis, a comprehensive study was carried out on the gear shaving process. Several commonly used methods for improving the tooth profile mid-concave error were also thoroughly discussed. The main conclusions obtained are as follows:
Firstly, a detailed analytical study of the gear shaving process was conducted, yielding fundamental conclusions such as the meshing equations, meshing line equations, contact paths, and the variation laws of contact points. The variations in relative velocity, induced normal curvature, and forces during the shaving process were derived in detail, elucidating the mechanisms by which they各自induce mid-concave errors.
Secondly, parametric modeling of the shaving cutter and the workpiece gear was performed, followed by finite element analysis. The results indicated that the two-point contact area located in the middle of the workpiece gear experienced the highest contact stress, quantitatively explaining the formation mechanism of the mid-concave error in gear shaving.
Thirdly, the principles and calculation formulas of negative-offset shaving and balanced shaving were elaborated and derived in detail. Experimental validation demonstrated that both methods can significantly improve the mid-concave error, albeit with their own limitations.
Fourthly, calculations of deformations during the shaving process were conducted using material mechanics and elastic mechanics. The obtained deformation values were numerically fitted to obtain a deformation curve, which is of significant reference value for formulating the modification curve of the shaving cutter. Additionally, a comparative analysis of deformations under different radial forces was conducted, revealing that the deformation increases significantly with increasing radial force, with the maximum deformation reaching approximately 0.012mm.