Abstract
Spiral bevel gear, as critical components in automotive transmission systems, play a significant role in terms of load-bearing capacity, transmission performance, high overlap coefficient, and low transmission noise. They are widely used in automotive manufacturing and high-end machine tools. Currently, gears are mainly produced through cutting, die forging, and ring rolling processes. Traditional gear cutting processes disrupt the material flow structure, have low production efficiency, and do not align with the concept of green manufacturing. In contrast, precision forging, as a near-net-shape technology, offers high processing accuracy, high production efficiency, and good material flow properties, making it suitable for manufacturing complex parts with high comprehensive performance requirements. Cold forging processes materials at room temperature, resulting in high deformation forces, while hot forging processes materials above the recrystallization temperature, prone to oxide scale formation on the forged surface. Warm forging, positioned between cold and hot forging, enhances mold life, reduces forming pressures, and improves forging precision.
This study utilizes Solidworks to establish a digital model of a spiral bevel gear and simulates the warm forging process using the Simulation module. Orthogonal experimental methods are employed to analyze the maximum stress and displacement of the gear under various billet temperatures and ram loads, thereby determining the optimal parameter set. The deformation of the tooth surface during warm forging of the spiral bevel gear is also investigated.
1. Introduction
Spiral bevel gears are key components in automotive transmission systems, known for their advantages such as good load-bearing capacity, strong transmission performance, high overlap coefficient, and low transmission noise. They are widely applied in automotive manufacturing and high-end machine tools . Currently, gears are primarily produced using cutting, die forging, and ring rolling processes . Traditional gear cutting processes disrupt the material flow structure, leading to low production efficiency and non-compliance with the concept of green manufacturing in China . In contrast, precision forging, as a near-net-shape technology, offers high processing accuracy, high production efficiency, and good material flow properties, making it suitable for manufacturing parts with complex shapes and high comprehensive performance requirements . Cold forging processes materials at room temperature, resulting in high deformation forces, while hot forging processes materials above the recrystallization temperature, which is prone to oxide scale formation on the forged surface. Warm forging, positioned between cold and hot forging, is considered a “new technology” that enhances mold life, reduces forming pressures, and improves forging precision .
Zhang Zaiping conducted a warm forging analysis of a geared shaft using numerical simulation methods and improved the structure of the forging mold based on this analysis. Cao Guoying carried out a warm forging analysis of the drive gear shaft of an automotive transmission, investigating the influence of process parameters such as initial billet temperature and friction between the billet and mold on the forming process. Huang Sitao et al. utilized Deform-3D technology to conduct numerical simulations on the precision forging of straight cylindrical gears, exploring the influence of parameters such as the location of the connecting skin, punch speed, and die structure on the forming process. Zhang Weiwei found through simulation of the warm forging of gear shafts that the bidirectional loading method of the mold is more conducive to forging. Qi et al. studied the warm forging of asymmetric gears and proposed a radial warm forging process to improve tooth angle filling. Hong-Seok Park et al. used a genetic algorithm to solve the multi-objective optimization problem of helical gear warm forging preform design. Yan et al. studied the warm forging process of large modulus gears through the finite element method and improved gear angle filling using a free-floating die. In summary, research by domestic and foreign scholars in the field of gear manufacturing has shown that the warm forging method is significant in improving gear production efficiency and product quality.
This study employs Solidworks to establish a digital model of a spiral bevel gear and simulates the warm forging process using the Simulation module. Orthogonal experimental methods are used to analyze the maximum stress and displacement of the gear under various billet temperatures and ram loads, determining the optimal parameter set. The deformation of the tooth surface during the warm forging of the spiral bevel gear is also investigated.
2. Establishment of the Spiral Bevel Gear Model
2.1 Material Properties
The material selected for the spiral bevel gear in this study is X38Cr-MoV5-3 alloy steel, known for its high toughness, wear resistance, thermal fatigue resistance, and minimal heat treatment distortion. The material properties are shown in Table 1.
Table 1. Material Property Parameters
| Property | Value |
|---|---|
| Elastic Modulus (E) | 215000 N/mm² |
| Poisson’s Ratio (ν) | 0.28 |
| Hardness (HRC) | 50-54 |
| Tensile Strength (σb) | 2000 MPa |
| Yield Strength (σs) | 1800 MPa |
| Thermal Expansion Coefficient (α) | 1.1×10⁻⁵ 1/°C |
| Elongation (%) | 10-14 |
2.2 Geometric Parameters and Finite Element Model
During transmission, short teeth are typically used to avoid tooth interference in spiral bevel gears. According to Gleason’s recommendations, when the number of teeth (Z1) of the pinion is ≥12, the working tooth height factor (fk) is 1.70, and the full tooth height factor (ft) is 1.888. The formulas for calculating the working tooth height (hk) and full tooth height (ht) of the spiral bevel gear are:
hk = 1.70m, ht = 1.888m (1)
The addendum height formulas for the gear teeth are:
hae1 = 1/2hk + x1m, hae2 = 1/2hk + x2m (2)
The dedendum height formulas are:
hfe1 = ht – hae1, hfe2 = ht – hae2 (3)
The root cone angles (θf1 for the pinion and θf2 for the gear) are given by:
tgθf1 = hfe1 / Re, tgθf2 = hfe2 / Re (4)
where Re is the outer cone distance.
The crown tip distance (Xe) is the distance from the crown (de) along the gear axis to the vertex of the pitch cone:
de1 = d1 + 2hae1cosδ1, de2 = d2 + 2hae2cosδ2 (5)
Xe1 = Re cosδ1 – hae1sinδ1, Xe2 = Re cosδ2 – hae2sinδ2 (6)
Based on the above gear parameter calculation formulas, the numerical values for the spiral bevel gear are obtained, as shown in Table 2.
Table 2. Spiral Bevel Gear Dimensional Parameters
| Parameter | Pinion Value | Gear Value |
|---|---|---|
| Module (m) | 2.5 | 2.5 |
| Number of Teeth (z) | 20 | 40 |
| Pitch Diameter (d) | 50.8 | 101.6 |
| Helix Angle (β) | 35 | 35 |
| Pressure Angle (α) | 20 | 20 |
| Crown Tip Distance (Xe) | 38.915 | 49.004 |
| Addendum Height (hae) | 3.018 | 2.540 |
| Dedendum Height (hfe) | 2.762 | 5.336 |
Using the dimensional parameters in Table 2, digital models of the large and small spiral bevel gears were established using Solidworks, as shown in Figure 1(a) and (b). The Simulation analysis module was used to create a finite element model of the spiral bevel gear, with an absolute mesh setting, a mesh tolerance of 0.1 mm, element size of 2.0 mm, and a total of 71961 elements, as shown in Figure 1(c).
Figure 1. Digital and Finite Element Models of the Spiral Bevel Gear
(a) Pinion Digital Model
(b) Gear Digital Model
(c) Finite Element Model
3. Warm Forging Simulation Analysis
3.1 Numerical Simulation Scheme
Based on the warm forging temperature range of 550-700°C for the forging material, an interval of 50°C was set. During warm forging, an excessively high ram load increases the material strength requirements, potentially causing mold damage, while an excessively low ram load results in incomplete tooth surface forming and inadequate performance. Therefore, the ram load was set to a range of 1000-4000 N, with an interval of 1000 N, as shown in Table 3.
Table 3. Orthogonal Experiment Design
| Sequence | Billet Temperature (°C) | Ram Load (N) | Maximum Stress (MPa) | Maximum Displacement (mm) |
|---|---|---|---|---|
| L1 | 550 | 1000 | 1387 | 0.0813 |
| L2 | 550 | 2000 | 1436 | 0.1345 |
| L3 | 550 | 3000 | 1371 | 0.1917 |
| L4 | 550 | 4000 | 1766 | 0.2464 |
| L5 | 600 | 1000 | 1479 | 0.0823 |
| L6 | 600 | 2000 | 1500 | 0.1368 |
| L7 | 600 | 3000 | 1537 | 0.1937 |
| L8 | 600 | 4000 | 1548 | 0.2499 |
| L9 | 650 | 1000 | 1624 | 0.0878 |
| L10 | 650 | 2000 | 1591 | 0.1387 |
| L11 | 650 | 3000 | 1727 | 0.1969 |
| L12 | 650 | 4000 | 1667 | 0.2522 |
| L13 | 700 | 1000 | 1778 | 0.0900 |
| L14 | 700 | 2000 | 1756 | 0.1424 |
| L15 | 700 | 3000 | 1761 | 0.1990 |
| L16 | 700 | 4000 | 1785 | 0.2545 |
3.2 Orthogonal Experiment Result Analysis
Simulations were conducted based on the set data. For a ram load of 4000 N, the maximum stresses on the gear tooth surface at forging temperatures of 550°C, 600°C, 650°C, and 700°C were 1766 MPa, 1548 MPa, 1667 MPa, and 1785 MPa, respectively. The maximum displacements were 0.2464 mm, 0.2499 mm, 0.2522 mm, and 0.2545 mm, respectively, as shown in Figure 2. The data on billet temperature, ram load, maximum stress, and maximum displacement are presented in Table 3.
From the simulation data in Table 3, the optimal warm forging parameters that reduce the maximum stress on the tooth surface of the spiral bevel gear while achieving a larger deformation displacement are a billet temperature of 600°C and a ram load of 4000 N. Under these conditions, the maximum stress on the spiral bevel gear is 1548 MPa, and the maximum displacement is 0.2499 mm, as shown in Figure 3.
Figure 2. Von Mises Stress and Maximum Displacement at Different Forging Temperatures with a Ram Load of 4000 N
(a) Stress at 550°C
(b) Stress at 600°C
(c) Stress at 650°C
(d) Stress at 700°C
(e) Displacement at 550°C
(f) Displacement at 600°C
(g) Displacement at 650°C
(h) Displacement at 700°C
Figure 3. Maximum Stress and Displacement at a Ram Load of 4000 N and Billet Temperature of 600°C
3.3 Temperature Field Analysis
During the warm forging process of the gear, the initial billet temperature is higher than the mold temperature, creating a temperature difference. As forging progresses, the contact area between the billet and the mold gradually increases, leading to heat exchange. This results in a temperature gradient between the surface and the core of the billet, causing the gear edges to protrude, as shown in Figure 4. As forging continues, the area with high central temperature in the gear gradually decreases, and the temperature difference between the forging and the mold narrows. The tooth edge is deformed under the ram load, generating significant heat, which increases the temperature of the tooth profile area and deforms the initially protruding edges, eventually forming the tooth surface. During the warm forging of the spiral bevel gear, the temperature field distribution is not uniform and changes with time and the forging die cavity space. As the billet temperature increases, the maximum stress on the gear tooth surface also increases, and the uneven temperature field distribution significantly affects the tooth surface forming process.
Figure 4. Temperature Field and Displacement Cloud at 600°C
4. Contact Analysis of the Spiral Bevel Gear Tooth Surface
The formula for the contact stress on the tooth surface of a spiral bevel gear is:
σH = ZEKAKVKHβZXZXC × 2000T1bd2e1ZI (7)
where:
- ZE is the elastic coefficient;
- KA is the overload coefficient;
- KV is the dynamic load coefficient;
- KHβ is the contact strength calculation load distribution coefficient;
- ZX is the size coefficient;
- ZI is the quality coefficient;
- ZXC is the crown coefficient.
The permissible contact stress is calculated as:
σHP = σHlimZNTZWSHKθZZ (8)
where:
- ZNT is the stress cycle coefficient;
- ZW is the hardness ratio coefficient;
- Kθ is the temperature coefficient.
The actual contact stress on the gear tooth surface must be less than or equal to the permissible contact stress, i.e., σH ≤ σHP (9).
The values of the coefficients in the contact stress formula are shown in Table 4.
Table 4. Parameters for Calculating Contact Stress of the Spiral Bevel Gear
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Elastic Coefficient (ZE) | 173.8 N/mm² | Stress Cycle Coefficient (ZNT) | 1.87 |
| Overload Coefficient (KA) | 1.0 | Hardness Ratio Coefficient (ZW) | 1.0 |
| Dynamic Load Coefficient (KV) | 1.0 | Temperature Coefficient (Kθ) | 0.75 |
| Load Distribution Coefficient (KHβ) | 1.4 | Size Coefficient (ZX) | 3.1 |
| Quality Coefficient (ZI) | 1.0 | Crown Coefficient (ZXC) | 0.8 |
For an input power (P) of 11 kW and a pinion speed (n1) of 970 r/min, the torque (T1) is calculated using Equation (10) as 108 N·m. Substituting the torque and coefficient values into Equation (7) yields a contact stress (σH) of 1631 MPa.
T1 = 9.55 × 106 × Pn1 (10)
5.Conclusion
To further evaluate the performance of the spiral bevel gear under the optimal warm forging conditions determined in the previous section, a contact stress field analysis was conducted. A torque of 108 N·m was applied to the pinion gear, and the friction coefficient was set to 0.15. The Simulation module in Solidworks was employed to simulate the contact stress distribution within the gear mesh.
The simulation results revealed that the maximum contact stress at the gear mesh interface was 510 MPa, as illustrated in Figure 5. This value is significantly lower than the allowable contact stress of 1631 MPa, which was calculated based on the gear parameters and operational conditions using the appropriate contact stress formula. The lower maximum contact stress indicates that the spiral bevel gear, forged under the conditions of 600°C and 4000 N, is capable of withstanding the applied load without exceeding the stress limits, ensuring its reliability and durability in practical applications.
The contact stress distribution map (Figure 5) provides a visual representation of the stress concentrations and their locations within the gear mesh. It can be observed that the stress levels gradually decrease away from the mesh interface, indicating a uniform stress distribution that contributes to the overall performance and lifespan of the gear.
In summary, the contact stress field analysis confirms that the spiral bevel gear, forged using the optimal parameters identified through simulation, exhibits satisfactory stress characteristics. The maximum contact stress is well within the safe limits, demonstrating the effectiveness of the warm forging process in producing high-quality gears suitable for demanding automotive applications.