Abstract:
Taking the concave problem of gear shaving as the research object, this paper carries out the deformation amount calculation of shaving gear and fitting analysis of modification curve through theoretical and experimental methods. A simplified mechanical model is used to analyze the forces under conditions of different numbers of contact points. Based on the theories of elasticity and material mechanics, the transient contact and bending deformation are calculated respectively, and then the total deformation is obtained. A modification method of shaving cutter for improving the concave problem is proposed, and a modification fitting curve is also worked out based on the results of deformation process by the least square method. The conclusion has reference value for improving the concave problem using the modification approach of shaving cutter.
1. Introduction
Gear transmission is a transmission mechanism used to transmit motion and power between any two shafts, featuring compact structure, smooth transmission, high efficiency, strong bearing capacity, and long service life. To meet the requirements of modern products for transmission performance, engineers have conducted relevant improvement research on gear tooth surface design, material selection, and surface treatment. The finishing process of the tooth surface has a significant impact on the transmission performance of the formed gear teeth. Gear shaving is an important part of the gear finishing process, which can be used as the final process of tooth profile processing to correct radial runout error, tooth pitch error, tooth profile error, and tooth orientation error of the gear ring, thereby improving the operational smoothness and contact strength of the shaved gear.
The process of gear shaving can be regarded as the meshing of a pair of non-side-clearance staggered-axis helical gears. During the shaving process, when a standard involute helical shaving cutter is used for shaving, the tooth profile of the shaved gear will produce varying degrees of concavity near the pitch circle, which is known as the shaving concave error. The concave error can lead to issues such as high noise and short lifespan in gear transmission. How to eliminate or reduce the shaving concave error has always been a major research focus for technicians. By analyzing the contact and bending deformations during the processing, a modification method for improving the concave error is proposed, and the modification curve is fitted based on numerical methods. Relevant research has important theoretical and practical value for the analysis of shaving processing accuracy and actual gear production.
2. Analysis of Gear Shaving Deformation
2.1 Contact Deformation During Gear Shaving
According to Hertz’s contact theory, the value of the tooth surface indentation, which is the contact deformation δe, during gear shaving is:
For the concave situation in gear shaving, it mostly occurs when the contact ratio is less than 2. Therefore, when analyzing the force at the contact points, only the case where the total number of contact points falls within the interval [2, 4] is considered. The following equations are established to solve the tooth surface force conditions when the number of contact points is 2, 3, and 4, respectively.
(Equations for different numbers of contact points are omitted for brevity but can be provided if needed.)
By solving the above equations, the contact deformation corresponding to each contact point can be further obtained, and subsequent detailed analysis of the shaving meshing situation can be conducted.
2.2 Bending Deformation During Gear Shaving
Due to the complexity of accurately calculating the bending deformation of helical gears, the shaved gear is approximately treated as a spur gear for bending deformation calculations. When the gear tooth is only subjected to one force, its variable-section cantilever beam model.
Using the segmented superposition method, the bending deformation at the contact point can be obtained.
Taking the width of the shaved gear as B, the bending deformation δwi caused by the shaded part in the y-direction is calculated. According to Castigliano’s theorem, the total bending deformation δw of point A under the action of FA along the y-direction can be expressed as:
When there are two contact points on the gear tooth, there are two cases, and their cantilever beam models.
Based on the theory of materials mechanics, bending deformations are calculated and analyzed separately. The bending deformation of point A can be expressed as:
The bending deformation of point A can be expressed as:
Through the calculation of contact and bending deformations, the total deformation δ of the shaved gear can be obtained:
3. Modification Method for Improving the Concave Problem in Gear Shaving
Currently, commonly used methods to improve the concave problem include the shaving cutter modification method, balanced shaving method, negative modification shaving method, small or large meshing angle shaving method, and changing the shaving process method. Extensive theoretical analysis and production practice have shown that the correct modification of the shaving cutter is an effective way to improve the shaving concave phenomenon. The so-called tooth profile modification of the shaving cutter refers to consciously modifying the shaving cutter’s pitch circle position to be concave to compensate for the defect of the concave tooth profile formed on the shaved gear during shaving. How to determine the modification position and amount, or whether a modification curve that can be controlled by parameters can be proposed to achieve the above purpose, this section uses numerical analysis methods to process the data to provide a reference for further detailed research.
The theoretical calculations of contact and bending deformations have been conducted. According to the calculation formulas, relevant data processing is performed using numerical analysis software Mathematica 6.0, and the least squares method is used to fit the results, providing a technical reference for shaving cutter modification. The normal equation for a quadratic polynomial is:
(The equation for the quadratic polynomial is omitted for brevity but can be provided if needed.)
After solving for a0, a1, and a2, the quadratic fitting curve can be obtained:
Given the parameters of the shaving cutter and the gear to be processed, as shown in Tables 1 and 2, respectively.
Table 1: Parameters of the Shaving Cutter
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Normal Module (mm) | 3 | Tip Diameter (mm) | 231.7 |
| Pressure Angle (°) | 20 | Modification Coefficient | -0.17 |
| Number of Teeth | 73 | Tooth Width (mm) | 20 |
| Helix Angle (°) | 15 | Radial Force (N) | 750 |
The normal module of the gear is 3 mm, indicating the standard size of the teeth. The tip diameter, which is the diameter at the outermost point of the teeth, is 45.231 mm. The pressure angle, a crucial parameter affecting the gear’s performance, is set at 20°, a common value for many gear applications. The modification coefficient, which is used to adjust the tooth profile for better performance, has a value of 0.3094 in this case.
The gear has 11 teeth, and the tooth width, which is the measurement across the face of the gear, is 15 mm. The helix angle, which is the angle between the teeth and the axis of the gear, is 28°, contributing to the gear’s ability to transmit torque efficiently. Finally, the radial force, representing the force acting perpendicular to the gear’s axis during operation, is 750 N.
These parameters are essential for understanding the gear’s performance and for designing and manufacturing processes, such as shaving, which aims to achieve precise tooth profiles and improve gear accuracy. The study referenced in the PDF focuses on calculating deformation during the shaving process and fitting a modification curve to the shaving cutter to address the concave problem, which can lead to issues like high noise and reduced lifespan in gear transmissions.