Abstract
Gear shaving is a widely applied precision finishing method for gear surfaces, significantly enhancing surface quality. This paper delves into the core factor influencing the surface quality of workpiece gears during gear shaving: the meshing contact characteristic. By considering the gear shaving process, mathematical and mechanical models for gear shaving are established based on meshing principles. A meshing contact analysis method for gear shaving is proposed, and the characteristics of gear shaving meshing contact under different contact ratios are obtained. The gear shaving tooth contact analysis is validated by the Finite Element Method (FEM). The results indicate that the maximum normal contact force on the tooth surface occurs near the pitch circle. As the contact ratio decreases, the normal contact force on the gear tooth surface increases, resulting in greater mesh impact and profile mid-concave errors.
1. Introduction
Gear shaving is a critical process in gear manufacturing, particularly for improving the surface quality of gear teeth. The contact ratio, a significant parameter in gear shaving, is believed to be a primary factor influencing the tooth profile mid-concave error. During the gear shaving process, the instantaneous meshing between the shaving cutter and the workpiece gear is point contact. Since the actual contact ratio is often not an integer, mutations in the number of contact points occur when the workpiece gear meshes near the pitch circle, leading to unbalanced contacts on the left and right meshing lines. This imbalance increases the cutting force acting on the gear tooth surface, resulting in excessive material removal and the formation of a prominent tooth profile mid-concave error.
2. Theoretical Background and Methodology
2.1 Gear Shaving Process and Contact Mechanics
The gear shaving process involves the continuous cutting of the gear tooth surface using a shaving cutter. When the meshing falls on the cutting edges at both ends of the chip flute, the normal velocities (v1n, v2n) of the contact points on the shaving cutter and workpiece gear are unequal, allowing the cutting edge to cut into the gear tooth surface. Simultaneously, the tangential velocities (v1t, v2t) are also unequal, resulting in relative slip between the tooth surfaces and forming the shaving action. The relative velocity vt(12) = v1t – v2t represents the shaving speed. The axial reciprocating motion of the workpiece gear ensures complete shaving of the target tooth surface.
2.2 Mathematical and Mechanical Models
Based on classical meshing principles, the meshing contact points satisfy the meshing equation: v(12) · n = 0, where v(12) is the relative motion velocity of the two tooth surfaces at the meshing point, and n is the normal vector at the meshing point. The meshing surface equation and tooth surface equation are derived using differential geometry.
2.3 Contact Analysis Algorithm
The tooth profile meshing points are determined by intersecting the tooth surface equation of the workpiece gear, projected onto the plane, with the meshing line equation. The normal forces, contact stresses, and deformations are then calculated using static equations and elastic contact deformation formulas.
3. Contact Characteristics Analysis
3.1 Contact Stress and Deformation
The contact characteristics of gear shaving meshing primarily investigate the contact stresses and deformations between the tooth surfaces of the shaving cutter and workpiece gear, including normal contact force (Fn), contact stress (σH), and contact deformation (δE). During the gear shaving process, the workpiece gear is simultaneously subjected to contact stress and bending stress on the tooth surface. However, bending deformation has a minor impact on the tooth surface and is therefore neglected.
3.2 Contact Ratio
The contact ratio indicates the number of tooth pairs simultaneously engaged during gear meshing. For spiral involute surfaces in staggered-axis meshing, point contact is used to increase the surface contact stress during gear shaving, enhancing the cutting effect. The contact ratio for gear shaving is given by:
ε’n = l + Δl / (πmn cos αn cos βb1)
where l is the effective length of end-face meshing, and Δl is the overrunning amount of the effective length of end-face meshing during shaving.
3.3 Calculation of Meshing Contact Characteristics
Based on the design principles and methods of shaving cutters, four shaving cutters were designed for different radial feeds, and four sets of gear shaving meshing models were established. The normal contact force, contact stress, and deformation curves were obtained through meshing contact analysis. The periodic variation of contact points for all four sets of models followed a 3-4-3-2-3-4 pattern.
Table 1. Basic Material Parameters of Workpiece and Shaving Cutter
| Parameter | Shaving Cutter (W18Cr4V) | Workpiece Gear (20CrMnTi) |
|---|---|---|
| Density (kg/m³) | 7800 | 7800 |
| Poisson’s Ratio | 0.3 | 0.25 |
| Young’s Modulus (MPa) | 218000 | 206000 |
Table 2. Basic Design Parameters of Shaving Cutters
| Parameter | Cutter 1 | Cutter 2 | Cutter 3 | Cutter 4 | Workpiece Gear |
|---|---|---|---|---|---|
| Number of Teeth | 53 | 52 | 53 | 52 | 17 |
| Module | 15° | 15° | 10° | 10° | 4.2333 |
| Pressure Angle | -0.3793 | -0.3744 | -0.3649 | -0.3603 | 20° |
| Helix Angle | 1.8294 | 1.7712 | 1.7133 | 1.6548 | -0.0468 |
| Modification Coefficient | – | – | – | – | – |
| Design Contact Ratio | – | – | – | – | – |
The results indicate that the normal contact force, contact stress, and deformation on the tooth surface exhibit clear step characteristics due to mutations in the meshing state and the number of contact points. The DE segment represents two-point contact, where under constant radial force, the normal contact force and deformation borne by the contact points on the tooth surface increase. The CD segment, representing three-point contact, shows similar forces and deformations as the DE segment due to unbalanced single-point contact forces, leading to increased susceptibility to plastic deformation.
The contact stress and strain trends, with comparable values, suggesting that further increasing the contact ratio when it is sufficiently large has minimal impact on tooth surface contact stress. Conversely, it can decrease the gear shaving center distance, potentially causing undercuts and root cuts. Comparing the analysis results for the four different contact ratios, it is evident that models with smaller contact ratios exhibit larger normal contact forces, contact stresses, and deformations near the pitch circle, increasing the likelihood of plastic deformation. As errors continually reflect, they ultimately manifest as noticeable tooth profile mid-concave errors near the pitch circle.
The finite element analysis (FEA) results show that the contact stress varies significantly under different meshing states. Initially, the contact stress is significantly lower than the theoretical values due to the transient process of reaching a steady state during mesh-in and meshing state changes. The stress curves for the four gear shaving models indicate that the stress near the middle of the tooth, primarily in the three-point meshing zone, tends to stabilize. Taking the average stress in this zone, the values are 618.076N, 622.400N, 665.865N, and 681.963N, respectively. This demonstrates that as the contact ratio decreases, the maximum stress during meshing increases nonlinearly.
4. Conclusions
This paper proposes a gear shaving meshing contact characteristic analysis method considering the existence of the chip groove and cutting edge on the shaving cutter gear. The theoretical research is verified through finite element simulations, and the main conclusions are as follows:
(1) The gear shaving meshing contact analysis reveals that the lower the contact ratio during gear shaving, the greater the maximum normal contact force that appears in the three-point meshing area near the pitch circle. Consequently, the shaving amount and deformation in this area also increase, making it more prone to generating the “mid-concave” error in the tooth profile.
(2) In gear shaving meshing, it is not the case that the larger the contact ratio, the better. When the contact ratio is sufficiently large, continuing to increase it has a minor impact on the shaving meshing contact performance. Conversely, excessively large contact ratios can lead to phenomena such as undercutting and root cutting.
(3) Finite element calculations indicate that the meshing contact stress tends to be stable near the pitch circle. The maximum meshing contact stress occurs in the three-point meshing area near the pitch circle. The maximum stress value calculated by the finite element method is slightly larger than that obtained through the gear shaving meshing contact analysis method, but it falls within the acceptable error range.
In summary, this paper provides valuable insights into the influence of contact ratio on the meshing contact characteristics during gear shaving. The proposed analysis method and findings can serve as a reference for optimizing the gear shaving process and improving the surface quality of workpiece gears.