Abstract:
Gear shaving is a critical finishing process in gear manufacturing, particularly in the automotive industry. However, the main issue during gear shaving is the occurrence of tooth profile distortion, known as the “concave phenomenon,” near the pitch circle of the shaved gear. To address this, a method for calculating the topographic modification of radial shaving cutters is proposed in this paper. By analyzing the geometric relationship between the shaving cutter and the shaved gear, a model for the modified tooth surface of the shaving cutter is established. This paper presents a detailed calculation and analysis of the modification amount on the radial shaving cutter’s tooth surface, aiming to provide a theoretical basis for the design and manufacture of similar radial shaving cutters.
1. Introduction
Radial gear shaving and gear grinding are advanced precision processing techniques for gears. Radial gear shaving has developed maturely abroad and is widely used in the automotive gear manufacturing industry. Various scholars have achieved certain results in the design and grinding of radial shaving cutter tooth surfaces. Previous studies mainly focused on the calculation of shaving cutter tooth profiles, grinding methods, and the arrangement of cutting grooves. However, these studies often used traditional calculation methods to obtain tooth profile modification curves, which were mostly two-dimensional and unable to fully reflect the three-dimensional surface of the entire tooth surface of the shaving cutter. As a result, accurate tooth surface modification during gear shaving was challenging. Therefore, this paper proposes a method to establish a model for the modified tooth surface of a radial shaving cutter and analyze its modification amount in detail.
2. Design of Modified Tooth Surface and Generation Equation
2.1 Design of Modified Tooth Surface of the Workpiece
The modified tooth surface of the gear differs from the theoretical tooth surface. It can be constructed by superimposing a theoretical tooth surface with a modification surface. The position vector and normal vector are expressed as follows:
R1r(u1,l1)=δ(u1,l1)n1(u1,l1)+R1(u1,l1)
N1r=(∂u1∂R1r×∂l1∂R1r)/∂u1∂R1r×∂l1∂R1r
Where R1 and n1 are the position vector and unit normal vector of the theoretical tooth surface of the gear, respectively; R1r and N1r are the position vector and normal vector of the modified tooth surface of the gear, respectively; δ is the normal modification amount; u1 and l1 are the parameters of the involute tooth surface.
2.2 Generation of Theoretical Tooth Surface of Radial Shaving Cutter
The principle of radial gear shaving is similar to the transmission of a pair of offset involute cylindrical gears. The helix angle of the shaving cutter and the helix angle of the workpiece form an axis intersection angle. The shaving cutter only moves in the radial direction of the workpiece with a small feed stroke. The tooth surface of the shaving cutter is the conjugate tooth surface that meshes with the involute helical surface of the workpiece in a crossed-axis manner. The envelope surface calculation is as follows:
Rs(u1,l1,θ1)=Mst(θs(θ1))MtfMf1(θ1)R1r(u1,l1)
Where θ1 is the workpiece rotation angle, θs=iθ1 is the shaving cutter rotation angle, i=Np/Ns is the transmission ratio (Np, Ns are the number of teeth of the workpiece and shaving cutter, respectively), E is the center distance, γ is the axis intersection angle, Sf, S1 are the reference and motion coordinate systems of the workpiece, respectively, St, Ss are the reference coordinate systems and motion coordinate systems of the shaving cutter, respectively, and Mst, Mtf, Mf1, Lst, Ltf, Lf1 are the position vector and normal vector transformation matrices from the tooth surface of the workpiece to the tooth surface of the shaving cutter.
3. Calculation of Modification Amount of Radial Shaving Cutter and Contact Analysis of Shaved Tooth Surface
The tooth surface of the radial shaving cutter is no longer an involute helical surface, but it differs from the theoretical involute helical surface by only a few tens of micrometers. The normal modification amount can be expressed as:
δs=(Rs−Rss)⋅nss
Where δs is the normal modification amount of the shaving cutter, Rss and nss are the position vector and unit normal vector of the involute gear tooth surface with the same basic parameters as the shaving cutter, respectively.
To determine the modification amount, a reference point M on the involute tooth surface (along the middle of the tooth width and tooth height) is selected, and its position vector RM is rotated by θ0 to coincide with the position vector RNs of the reference point N on the tooth surface of the shaving cutter.
Based on the above, the modification amount of the radial shaving cutter can be calculated, and the instantaneous contact line distribution on the tooth surface of the workpiece can be obtained by simulating the meshing of the shaving cutter and the workpiece tooth surface.
4. Case Study and Analysis
To further analyze the modification amount of the radial shaving cutter and the variation in contact of the shaved tooth surface under specific parameters, a computer simulation study was conducted on the influence of key parameters such as the number of teeth, pressure angle, and helix angle on the modification amount of the radial shaving cutter. Limited by space, the simulation results are analyzed using the basic parameters in Table 1 as an example.
Table 1. Basic Parameters
| Category | Number of Teeth | Module / (mm) | Pressure Angle / (°) | Helix Angle / (°) | Tooth Width / (mm) | Helical Direction |
|---|---|---|---|---|---|---|
| Workpiece | 18 | 1.75 | 20 | 5 | 20 | Right-handed |
| Shaving Cutter | 137 | 1.75 | 20 | 10 | 22 | Right-handed |
(1) Influence of Axis Intersection Angle and Number of Teeth of Shaving Cutter on Modification Amount:
- As the number of teeth of the shaving cutter increases, the modification amount gradually decreases. The tooth surface exhibits an anti-bulging twisted surface.
- As the helix angle (axis intersection angle) increases, the modification amount gradually increases. When the axis intersection angle is less than 10°, there is basically no tooth direction twisting phenomenon, mainly tooth profile modification. When the axis intersection angle is greater than 10°, tooth direction twisting occurs.
(2) Influence of Installation Error on Modification Amount of Radial Shaving Cutter:
- The tooth surface of the shaving cutter after indexing is not very sensitive to center distance errors.
- The tooth surface of the shaving cutter after indexing is sensitive to axis intersection angle errors.
(3) Variation of Modification Amount on the Tooth Surface of the Radial Shaving Cutter for Modified Gears:
- When the tooth profile of the workpiece is modified, the tooth surface of the shaving cutter basically exhibits a topological shape of tooth profile formation, and the tooth direction twisting phenomenon basically disappears.
- When the tooth direction of the workpiece is modified, the tooth surface of the shaving cutter still exhibits an anti-bulging twisted phenomenon, but the tooth direction twisting phenomenon basically disappears.
- After three-dimensional modification of the workpiece, the tooth surface of the shaving cutter basically exhibits a topological shape of tooth profile formation, and the tooth direction twisting phenomenon basically disappears.
(4) Simulation of Tooth Surface Contact Changes During Gear Shaving:
As the helix angle of the workpiece changes within the range of β = 0–15°, the corresponding axis intersection angle ranges from 10° to 25°. During the meshing process, the contact situation between the tooth surfaces of the shaving cutter and the workpiece is simulated. When the helix angle of the shaving cutter remains constant, and the helix angle of the workpiece varies within the range of β=0–15°, the corresponding axis intersection angle varies within the range of 10° to 25°.
That during the initial and final stages of meshing, the instantaneous contact line is relatively short, and the contact between the shaving cutter and the workpiece is linear, covering the entire tooth surface of the workpiece. Typically, during shaving, the contact ratio coefficient εa ranges from 2 to 3, with multiple teeth in contact at the initial and final stages of meshing, indicating a high contact ratio. If the contact ratio remains relatively stable throughout the shaving process, the tooth profile concavity phenomenon can be minimized. Therefore, when the axis intersection angle is less than 10°, the contact ratio changes more smoothly throughout the shaving process, which is beneficial for reducing the tooth profile concavity phenomenon.