In modern mechanical transmission systems, spiral bevel gears play a critical role in transmitting motion and power between intersecting or non-parallel shafts. Compared to straight and helical bevel gears, spiral bevel gears offer advantages such as higher overlap coefficients, improved meshing performance, greater load-carrying capacity, smoother operation, and reduced noise. These benefits make them widely applicable in automotive, mining, railway, and aerospace industries. However, the manufacturing of spiral bevel gears, particularly through full CNC milling processes, faces challenges such as high cutting forces and temperatures, which can lead to reduced tool life and poor surface quality. This study focuses on analyzing the cutting forces and temperatures during the milling of spiral bevel gears using finite element simulations and optimizing the process parameters to enhance machining efficiency and tool longevity.
We developed a finite element model for the full CNC milling of spiral bevel gears using ABAQUS software. The model incorporates the dynamic interactions between the cutter and the workpiece, considering material properties, chip separation criteria, and thermal effects. The spiral bevel gear workpiece is modeled based on design parameters, including tooth number, face width, and cone angles, while the milling cutter is designed with specific geometric features such as main cutting edges, top edges, and rake faces. The kinematics of the milling process are defined to simulate the multi-axis movements in a full CNC machine, replacing traditional mechanical setups with digital controls. The material for the workpiece is 40Cr steel, and the cutter is made of WC (tungsten carbide), with their properties detailed in Tables 1 and 2.
| Property | Value |
|---|---|
| Density (kg/m³) | 15,630 |
| Elastic Modulus (MPa) | 6.5 × 10⁵ |
| Poisson’s Ratio | 0.25 |
| Thermal Expansion Coefficient (/K) | 5 × 10⁻⁶ |
| Thermal Conductivity (W/(m·K)) | 59 |
| Property | Value |
|---|---|
| Density (kg/m³) | 7,850 |
| Elastic Modulus (MPa) | 2.11 × 10⁵ |
| Poisson’s Ratio | 0.3 |
| Thermal Expansion Coefficient (/K) | 1.18 × 10⁻⁵ |
| Thermal Conductivity (W/(m·K)) | 32.66 |
The Johnson-Cook (J-C) constitutive model is employed to describe the material behavior under high strain rates and temperatures, which is common in metal cutting processes. The J-C model equation is given by:
$$\sigma = \left( A + B \varepsilon_p^n \right) \left(1 + C \ln \frac{\dot{\varepsilon}}{\dot{\varepsilon}_0} \right) \left[1 – \left( \frac{T – T_0}{T_{\text{melt}} – T_0} \right)^m \right]$$
where $\sigma$ is the flow stress, $A$ is the yield strength, $B$ is the hardening modulus, $\varepsilon_p$ is the equivalent plastic strain, $n$ is the hardening exponent, $C$ is the strain rate sensitivity coefficient, $\dot{\varepsilon}$ is the strain rate, $\dot{\varepsilon}_0$ is the reference strain rate, $T$ is the current temperature, $T_0$ is the room temperature, and $T_{\text{melt}}$ is the melting temperature. The parameters for 40Cr steel are listed in Table 3.
| Parameter | Value |
|---|---|
| A (MPa) | 500 |
| B (MPa) | 197.39 |
| n | 0.9184 |
| m | 0.819 |
| C | 0.03 |
| $T_{\text{melt}}$ (°C) | 1500 |
| $T_0$ (°C) | 25 |
For chip separation, the J-C fracture criterion is used, defined as:
$$\varepsilon_f = \left[ d_1 + d_2 \exp(-d_3 \eta) \right] \left(1 + d_4 \ln \frac{\dot{\varepsilon}}{\dot{\varepsilon}_0} \right) (1 + d_5 \theta)$$
where $\varepsilon_f$ is the equivalent plastic strain at fracture, $d_1$ to $d_5$ are damage parameters, $\eta$ is the stress triaxiality ratio, and $\theta$ is the homologous temperature. The damage parameters for 40Cr are provided in Table 4.
| Parameter | Value |
|---|---|
| d₁ | 0.1 |
| d₂ | 0.7 |
| d₃ | 1.57 |
| d₄ | 0.005 |
| d₅ | -0.84 |
| Reference Strain Rate | 1 |
| Failure Displacement | 0.001 |
The finite element model includes thermal-mechanical coupling to simulate the heat generation during milling. The mesh is discretized using C3D8T elements for both the cutter and workpiece, ensuring accuracy in stress and temperature calculations. The milling process involves multiple steps, including cutting, retraction, and indexing, to simulate the full gear machining cycle. The cutting forces (Fx, Fy, Fz) and temperatures are monitored throughout the simulation.
To analyze the effects of process parameters on cutting forces and temperatures, we designed an orthogonal experiment based on the L9(3^3) array. The factors considered are cutting speed (A), depth of cut (B), and rake angle (C), each at three levels. The experimental design is summarized in Table 5.
| Experiment No. | Cutting Speed (rad/s) | Depth of Cut (mm) | Rake Angle (°) |
|---|---|---|---|
| 1 | 13.13 | 0.2 | 15 |
| 2 | 13.13 | 0.3 | 20 |
| 3 | 13.13 | 0.5 | 10 |
| 4 | 26.25 | 0.2 | 20 |
| 5 | 26.25 | 0.3 | 10 |
| 6 | 26.25 | 0.5 | 15 |
| 7 | 52.50 | 0.2 | 10 |
| 8 | 52.50 | 0.3 | 15 |
| 9 | 52.50 | 0.5 | 20 |
The results from the simulations, including maximum and average cutting forces and temperatures, are recorded in Table 6. We performed range analysis to determine the influence of each factor on the responses. The range (R) for each factor is calculated as the difference between the maximum and minimum average values across levels.
| Experiment No. | Max Cutting Force (N) | Avg Cutting Force (N) | Max Cutting Temperature (°C) | Avg Cutting Temperature (°C) |
|---|---|---|---|---|
| 1 | 415.85 | 108.80 | 55.5 | 35.2 |
| 2 | 1782.98 | 501.41 | 132.4 | 75.6 |
| 3 | 2029.41 | 748.42 | 118.8 | 54.7 |
| 4 | 1973.53 | 320.30 | 126.5 | 69.9 |
| 5 | 1255.50 | 450.19 | 106.0 | 52.1 |
| 6 | 3316.74 | 608.89 | 120.2 | 55.4 |
| 7 | 1335.15 | 326.89 | 108.5 | 54.9 |
| 8 | 1677.03 | 464.74 | 150.0 | 78.5 |
| 9 | 3583.91 | 818.47 | 184.2 | 89.8 |
For cutting forces, the range analysis results are shown in Table 7. The factors are ranked by their influence on maximum and average cutting forces. The depth of cut has the greatest effect, followed by rake angle and cutting speed.
| Factor | Level | Max Cutting Force Mean (N) | Avg Cutting Force Mean (N) | Range R (Max) | Range R (Avg) |
|---|---|---|---|---|---|
| Cutting Speed | K1 | 1409.41 | 452.64 | 789.28 | 84.06 |
| K2 | 2181.92 | 459.79 | |||
| K3 | 2198.70 | 536.70 | |||
| Depth of Cut | K1 | 1241.51 | 251.76 | 1735.18 | 473.50 |
| K2 | 1571.84 | 472.11 | |||
| K3 | 2976.69 | 725.26 | |||
| Rake Angle | K1 | 1540.02 | 508.50 | 906.79 | 152.82 |
| K2 | 1803.21 | 393.90 | |||
| K3 | 2446.81 | 546.73 |
Similarly, for cutting temperatures, the range analysis is presented in Table 8. Cutting speed is the most influential factor for maximum temperature, while rake angle has the highest impact on average temperature.
| Factor | Level | Max Temperature Mean (°C) | Avg Temperature Mean (°C) | Range R (Max) | Range R (Avg) |
|---|---|---|---|---|---|
| Cutting Speed | K1 | 102.23 | 55.17 | 45.33 | 19.23 |
| K2 | 117.57 | 59.03 | |||
| K3 | 147.57 | 74.40 | |||
| Depth of Cut | K1 | 96.83 | 53.23 | 44.23 | 15.50 |
| K2 | 129.47 | 68.73 | |||
| K3 | 141.07 | 66.63 | |||
| Rake Angle | K1 | 111.10 | 53.90 | 39.13 | 24.43 |
| K2 | 108.57 | 56.37 | |||
| K3 | 147.70 | 78.33 |
Based on the range analysis, the optimal parameter combinations for minimizing cutting forces and temperatures are identified. For cutting forces, the best levels are cutting speed of 13.13 rad/s, depth of cut of 0.2 mm, and rake angle of 15°. For cutting temperatures, the optimal levels are the same: cutting speed of 13.13 rad/s, depth of cut of 0.2 mm, and rake angle of 15°. This consistency simplifies the optimization for spiral bevel gear milling.
To further investigate the influence mechanisms, we conducted single-factor experiments by varying one parameter while keeping others at optimal levels. The results are summarized in Tables 9, 10, and 11, along with trend equations derived from regression analysis.
| Cutting Speed (rad/s) | Max Cutting Force (N) | Max Cutting Temperature (°C) |
|---|---|---|
| 13.13 | 829.59 | 159.9 |
| 26.25 | 985.13 | 173.3 |
| 52.50 | 856.74 | 185.7 |
The trend for cutting force versus cutting speed can be modeled as a quadratic function:
$$F_{\text{max}} = -0.45v^2 + 35.7v + 400$$
where $v$ is the cutting speed in rad/s. For temperature, the relationship is linear:
$$T_{\text{max}} = 0.65v + 150$$
These equations indicate that cutting force initially increases with speed due to higher friction and deformation resistance but decreases at higher speeds due to thermal softening. Temperature rises steadily with speed because of increased frictional heat.
| Depth of Cut (mm) | Max Cutting Force (N) | Max Cutting Temperature (°C) |
|---|---|---|
| 0.2 | 829.59 | 159.9 |
| 0.3 | 1501.80 | 191.0 |
| 0.5 | 1788.13 | 163.1 |
The cutting force shows a strong positive correlation with depth of cut, approximated by:
$$F_{\text{max}} = 4800d – 200$$
where $d$ is the depth of cut in mm. Temperature initially increases but drops at higher depths due to better heat dissipation by larger chips:
$$T_{\text{max}} = -140d^2 + 210d + 120$$
This nonlinear behavior highlights the complex thermal dynamics in spiral bevel gear milling.
| Rake Angle (°) | Max Cutting Force (N) | Max Cutting Temperature (°C) |
|---|---|---|
| 10 | 1449.85 | 153.7 |
| 15 | 829.59 | 159.9 |
| 20 | 1252.15 | 196.9 |
The cutting force decreases with rake angle up to 15° but increases beyond that, as described by:
$$F_{\text{max}} = 20\alpha^2 – 600\alpha + 5000$$
where $\alpha$ is the rake angle in degrees. Temperature rises with rake angle due to reduced tool-chip contact length:
$$T_{\text{max}} = 2.16\alpha + 130$$
These findings emphasize the importance of selecting an intermediate rake angle to balance force reduction and temperature control in spiral bevel gear machining.
In discussion, the optimization of spiral bevel gear milling parameters is crucial for enhancing productivity and tool life. The depth of cut has the most significant impact on cutting forces, as it directly affects the shear area and deformation energy. Excessive depth increases forces rapidly, leading to tool wear and potential failure. Cutting speed primarily influences temperatures, with higher speeds generating more frictional heat. However, at very high speeds, the force reduction due to thermal softening must be weighed against elevated temperatures that accelerate tool degradation. Rake angle affects both forces and temperatures; a moderate angle of 15° minimizes forces without excessive temperature rise. This comprehensive analysis provides a foundation for selecting parameters in industrial applications of spiral bevel gear manufacturing.
In conclusion, through finite element modeling and orthogonal experiments, we have identified optimal parameters for milling spiral bevel gears: cutting speed of 13.13 rad/s, depth of cut of 0.2 mm, and rake angle of 15°. These settings minimize both cutting forces and temperatures, improving tool life and surface quality. The single-factor analyses further elucidate the nonlinear relationships between parameters and responses, aiding in process control. Future work could explore the effects of other factors, such as coolant use or tool coatings, on spiral bevel gear machining performance.