Abstract: Exploring the influence of contact characteristics on the concave errors in the process of gear shaving is of great significance for understanding the formation mechanism of these errors. Based on the elastoplastic theory and Loaded Tooth Contact Analysis (LTCA), a mechanical model is constructed to analyze the meshing contact characteristics under different loading conditions. This paper elucidates the formation mechanism of concave errors. The finite element method (FEM) is applied to clarify the tooth surface contact stress and the division of elastoplastic deformation regions under various loads, which is compared with gear shaving experiments. The results indicate that the plastic deformation area on the tooth profile increases nonlinearly with the increase in load. The tooth root experiences greater stress and deformation than the tooth tip, with peak values occurring near the middle of the pitch circle, where plastic deformation is most likely. As low-cycle meshing occurs during gear shaving, plastic deformation accumulates, leading to the reproduction of tooth profile errors and ultimately resulting in a noticeable concave error phenomenon on the tooth profile. The contact stress and elastoplastic region division results obtained through finite element simulations are reliable, and the theoretical analysis and research conclusions are verified through experiments.
1. Introduction
The presence of concave errors in the tooth profile of shaved gears is one of the main factors causing vibration and noise in gear transmissions, significantly affecting gear lifespan and transmission performance. The gear shaving process involves complex staggered-axis spatial discontinuous surface meshing, accompanied by various comprehensive errors generated by the process system, which complicates the tooth surface forming process. Therefore, concave errors have become a worldwide problem. Scholars at home and abroad generally believe that the main reason for the occurrence of concave errors is the reduction in the number of meshing contact points near the pitch circle, causing an imbalance in left and right meshing line contacts, which results in impact forces in this area. This leads to an increase in the cutting depth of the shaving cutter, accompanied by the reproduction of random errors, thereby forming a middle concave depression of 0.01 to 0.03 mm.
2. Literature Review
Litvin et al. proposed Loaded Tooth Contact Analysis (LTCA) technology for spiral bevel gears, which is used to study the contact mechanical properties of gears. This technology introduces edge contact theory based on Tooth Contact Analysis (TCA), effectively overcoming the shortcomings of TCA. However, in the LTCA analysis process, when more than two gears are meshing, issues such as pressure distribution on the meshing tooth surfaces and the magnitude of tooth surface deformation cannot be resolved. These issues require further solution using methods such as Mohr’s energy method in material mechanics, elastic mechanics, and the finite element flexibility matrix method. Tang Jinyuan et al. proposed Tooth Error Contact Analysis (ETCA) based on the multi-body system error modeling theory, using the SGM method for bevel gear processing as an example. They obtained a quantitative relationship between machine tool motion errors and installation errors on the processing quality of tooth surfaces. Compared with TCA technology, ETCA analysis results provide more reasonable guidance for processing. Moriwaki et al. established a new random cutting model based on elastic theory to predict the cutting effect of the shaving cutter after finish processing and simulated the shaving process. They analyzed the meshing conditions and load distribution, verifying the good effect of small meshing angle shaving. Chen et al. established a meshing model for spiral gears and double-crown gears, using elastic theory and the finite element method to analyze the contact stress and bending stress of spiral gears, verifying the consistency of the two methods. However, during the shaving process, both elastic and plastic deformations coexist on the tooth surface. Studying concave errors in the gear tooth profile requires simultaneous consideration of the influence of elastoplastic deformation on tooth surface processing.
3. Methodology
3.1 Geometric Model
In gear shaving, due to the presence of chip grooves and cutting edges on the tooth surface of the shaving cutter, the contact line on the workpiece tooth surface is an intermittent spatial curve, complicating the meshing contact problem. According to literature , the shaving process can be simplified as a staggered-axis spatial meshing motion between a pair of helical-straight gears. Since most concave errors occur in cutter-tooth pairs with a shaving meshing contact ratio less than 2, the parameters of the shaving cutter and workpiece gear are selected as shown in Table 1, and corresponding shaving meshing parameters are derived, with a contact ratio of 1.401. A three-dimensional geometric model is established as shown in Figure 1, with corresponding coordinate systems established for subsequent mathematical model derivation.
Table 1. Parameters of Shaving Cutter and Work Gear
| Parameter | Shaving Cutter | Work Gear |
|---|---|---|
| Number of teeth | 43 | 12 |
| Module | 5.35 | 5.35 |
| Pressure angle | 20° | 20° |
| Helix angle | 11° | – |
| Arc tooth thickness at pitch circle | 6.6 | 10.54 |
| Material | W18Cr4V | 20CrMnTi |
| Density (kg/m³) | 7800 | 7800 |
| Poisson’s ratio | 0.3 | 0.25 |
| Young’s modulus (MPa) | 218000 | 206000 |
| Yield strength (MPa) | – | ≥835 |
Figure 1. Geometric Model of Shaving Meshing
3.2 Mathematical Model
Gear shaving is a two-degree-of-freedom meshing process. Based on the meshing principle and the parameters, the meshing equation is expressed using coordinate system S1 (O1-x1y1z1) as follows:
The geometric meaning is that when a certain rotation angle θ is selected, a series of points on the tooth surface Σ(1) of the shaving cutter simultaneously satisfy the meshing condition, meaning there is a meshing contact line on the tooth surface at this time. If both equations (2) and (3) need to be satisfied simultaneously, then only one pair of (u, θ) solutions exists, i.e., there is only one contact intersection point on the tooth surface Σ(1) of the shaving cutter at any instant. Obviously, if the forms of the two conditional equations are the same or correspond in proportion, it can be considered that the point contact of gear shaving meshing transforms into line contact, satisfying the following transformation condition:
(Equation related to the transformation condition is omitted here for brevity.)
After arranging equations (1), (2), and (3) and substituting them into the transformation condition equation, the two equations can be solved to obtain a unique solution in a simplified form.
3.3 Mechanical Model and LTCA
Gear shaving is actually a process of no-backlash meshing extrusion cutting. Taking the processing of the right tooth surface Σ(2) of the workpiece gear as an example, when the left (right) tooth surface of the shaving cutter is the cutting side, the right (left) tooth surface of the shaving cutter only serves for extrusion polishing and balancing torque. It can be seen that the left and right tooth surfaces of the workpiece gear are symmetrically stressed during a complete shaving process. Therefore, the right tooth profile of the workpiece gear is selected as the research object. Based on the above mathematical model, a mechanical model for gear shaving meshing is established, and Loaded Tooth Contact Analysis (LTCA) is performed on the model.
After solving the normal forces at the contact points during the meshing process, the corresponding elastic deformation compression δe can be obtained. The contact stress σH on the tooth surface is calculated using the AGMA standard formula for helical gear tooth surface contact stress.
By selecting four different radial forces (700N, 1100N, 1500N, and 2000N), the right tooth profile is analyzed for meshing contact using the above method. The normal forces in each state are obtained through static analysis, and then the deformation and contact stress values for any curvature radius of the right tooth profile are determined using equations (9) and (10). The meshing contact characteristic curves under different load conditions are obtained as shown in Figure 3.
Figure 3. Meshing Contact Characteristic Curves under Different Radial Forces
(Description and labels of Figure 3 are omitted here for brevity, but generally show the variation of contact points, contact stress, and deformation along the tooth profile under different radial forces.)
4. Elastoplastic Analysis
During the initial stage of gear shaving, the normal forces at the meshing contact points between the shaving cutter and the workpiece gear are relatively small, and the contact stress is within the elastic range. Upon completion of the shaving process, the elastic deformation will recover due to the disappearance of the applied forces. As the shaving process progresses, the radial force increases with the increase in feed rate, and consequently, the normal forces at the contact points also increase.
As illustrated in the relevant figures, when the radial force Fr reaches 1500N, the stress in the two-point contact segment CD, the three-point contact segment DE, and part of the four-point contact segment EF exceeds 835MPa, indicating that plastic deformation may occur in these areas.
The distinction between elastic and plastic deformation zones on the tooth profile is crucial for guiding the elimination of the effects of concave errors. As plastic deformation cannot recover and will be reflected as errors on the tooth surface after repeated shaving cycles, it ultimately leads to tooth surface errors upon completion of the shaving process. Therefore, accurately delineating the elastic and plastic deformation zones on the tooth profile holds significant importance for mitigating the impact of concave errors.