Involute spur cylindrical gears are widely used in mechanical transmission systems due to their high load-bearing capacity, efficiency, and cost-effectiveness. However, during long-term service, friction and wear on the tooth surfaces can lead to reduced transmission accuracy and efficiency. To address this, tooth profile modification, such as drum-shaped crowning, is often applied to restore the original tooth geometry and enhance performance. This study investigates the effects of drum-shaped tooth profile modification on the contact stress distribution and fatigue lifetime of parallel axis involute spur cylindrical gears using finite element analysis. We employ SolidWorks for geometric modeling and ANSYS for transient dynamics and fatigue simulations. The results demonstrate that appropriate modification can improve meshing performance, but excessive modification may induce stress concentration and reduce fatigue life. Additionally, under misalignment conditions, proper crowning can mitigate adverse effects, extending the gear’s service life.
The involute profile of spur gears ensures constant pressure angle during meshing, which contributes to smooth power transmission. The standard design parameters for spur gears include module, pressure angle, number of teeth, addendum coefficient, and dedendum coefficient. For this study, we consider a pair of spur gears with 40 teeth each, a module of 2 mm, and a pressure angle of 20°. The gear dimensions are calculated as follows: pitch diameter $d = m \times z = 80$ mm, base diameter $d_b = d \times \cos(20^\circ) = 75.18$ mm, addendum $h_a = h_a^* \times m = 2$ mm, dedendum $h_f = (h_a^* + c^*) \times m = 2.5$ mm, and total tooth height $h = h_a + h_f = 4.5$ mm. The gear width is set to 20 mm. Using SolidWorks, we generate the 3D geometric model of the spur gear pair based on the involute curve equations:
$$x_k = r_b \sin u_k – r_b u_k \cos u_k$$
$$y_k = r_b \cos u_k + r_b u_k \sin u_k$$
where $r_b$ is the base radius and $u_k$ is the sum of the roll angle and pressure angle. The tooth profile is extruded, and fillets are added at the root to reduce stress concentration.
Drum-shaped modification is applied to the tooth surfaces to optimize load distribution and reduce edge contact. This includes both profile crowning (along the tooth height) and lead crowning (along the tooth width). The modification follows a parabolic curve, defined as:
$$\Delta x = \Delta x_{\text{max}} \left\{ \frac{[y + (h_f – c)] – y_0}{y_0} \right\}^2, \quad – (h_f – c) \leq y \leq h_a$$
$$\Delta x = \Delta x_{\text{max}} \left[ \frac{(z + b_f) – z_0}{z_0} \right]^2, \quad – b_f \leq z \leq b_r$$
where $\Delta x$ is the modification amount in the tooth normal direction, $\Delta x_{\text{max}}$ is the maximum modification (denoted as $C_\alpha$ for profile crowning and $C_\beta$ for lead crowning), $y_0$ and $z_0$ are symmetry positions, and $b_f$ and $b_r$ are distances from the origin to the front and rear faces. We examine six modification cases, with $C_\alpha$ and $C_\beta$ ranging from 0 μm to 10 μm, as summarized in Table 1.
| Modification Case | Maximum Profile Modification $C_\alpha$ (μm) | Maximum Lead Modification $C_\beta$ (μm) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 2 | 2 |
| 3 | 4 | 4 |
| 4 | 6 | 6 |
| 5 | 8 | 8 |
| 6 | 10 | 10 |
For finite element analysis, we use ANSYS to perform transient dynamics and fatigue simulations. The spur gears are modeled with 40Cr alloy steel, which has a density of 7870 kg/m³, Young’s modulus of 211 GPa, Poisson’s ratio of 0.277, yield strength of 785 MPa, and ultimate tensile strength of 980 MPa. The gear pair is meshed with 10-node tetrahedral elements (Tet10), and the contact surfaces are refined to ensure accuracy. The number of elements and nodes for each modification case are listed in Table 2.
| Modification Case | Element Type | Number of Elements | Number of Nodes |
|---|---|---|---|
| 1 | Tet10 | 223961 | 383685 |
| 2 | Tet10 | 246221 | 419235 |
| 3 | Tet10 | 246458 | 419563 |
| 4 | Tet10 | 245773 | 418609 |
| 5 | Tet10 | 246392 | 419390 |
| 6 | Tet10 | 246660 | 419513 |
In the transient dynamics analysis, the driving spur gear is subjected to a rotational velocity of 2 rad/s, while the driven spur gear experiences a resisting torque of 15000 N·mm. The analysis time step is set to 1 s, divided into 20-250 sub-steps. Frictional contact with a coefficient of 0.15 is defined between all tooth surfaces. The results show that unmodified spur gears (Case 1) exhibit meshing interference, with stress concentration at the pitch line and tooth root, reaching a maximum von Mises stress of 107.37 MPa. With moderate modification (Case 2), the stress distribution improves, and the maximum stress decreases to 103.13 MPa. However, as modification increases, the contact area shrinks toward the tooth center, leading to higher stress levels—158.35 MPa for Case 3, 201.54 MPa for Case 4, 165.8 MPa for Case 5, and 247.52 MPa for Case 6. The contact stress over time for each case is plotted, revealing that higher modification amounts increase stress fluctuations, which may accelerate fatigue failure.
For fatigue lifetime analysis, we use the ANSYS Fatigue Tool module with the S-N curve of 40Cr steel, derived from experimental data. The S-N curve equation is:
$$N_f = 1.82524 \times 10^{11} (\sigma_m – 250)^{-2.12613}$$
where $N_f$ is the fatigue life in cycles, and $\sigma_m$ is the mean stress. The fatigue life results for ideal alignment (no misalignment) are presented in Table 3. Unmodified spur gears have the highest fatigue life of 23.733 million cycles. As modification increases, the fatigue life decreases: 19.936 million cycles for Case 2, 6.1046 million for Case 3, 2.2992 million for Case 4, 2.551 million for Case 5, and below 0.05 million for Case 6. This indicates that excessive modification reduces the fatigue resistance of spur gears due to stress concentration.
| Modification Case | Fatigue Life (cycles) |
|---|---|
| 1 | 23.733 × 10⁶ |
| 2 | 19.936 × 10⁶ |
| 3 | 6.1046 × 10⁶ |
| 4 | 2.2992 × 10⁶ |
| 5 | 2.551 × 10⁶ |
| 6 | < 0.05 × 10⁶ |
We also investigate the effects of axis misalignment on spur gear performance. Misalignment angles of 0.1°, 0.2°, 0.3°, and 0.4° are considered, with a reduced torque of 14000 N·mm to ensure convergence. Under misalignment, unmodified spur gears show significant stress concentration at the tooth edges, with maximum von Mises stress reaching 366.98 MPa at 0.2° misalignment. With modification, the stress distribution improves. For example, at 0.2° misalignment, the maximum stress decreases to 283.79 MPa for Case 2, 168.23 MPa for Case 3, 176.56 MPa for Case 4, 190.87 MPa for Case 5, and 189.94 MPa for Case 6. The contact area shifts toward the tooth center, reducing edge loading. The fatigue life under misalignment is summarized in Table 4. Unmodified spur gears are highly sensitive to misalignment, with life dropping sharply. In contrast, spur gears with higher modification (e.g., Cases 4-6) maintain relatively stable fatigue lives, demonstrating that drum-shaped modification can compensate for misalignment effects.
| Modification Case | Fatigue Life at 0.1° (cycles) | Fatigue Life at 0.2° (cycles) | Fatigue Life at 0.3° (cycles) | Fatigue Life at 0.4° (cycles) |
|---|---|---|---|---|
| 1 | ~10⁶ | ~10⁵ | ~10⁴ | ~10³ |
| 2 | ~25 × 10⁶ | ~25 × 10⁶ | ~10⁶ | < 0.1 × 10⁶ |
| 3 | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ | ~10⁶ |
| 4 | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ |
| 5 | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ |
| 6 | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ | ~25 × 10⁶ |
In conclusion, drum-shaped tooth profile modification significantly influences the performance and fatigue lifetime of involute spur cylindrical gears. Under ideal alignment, moderate modification improves meshing by reducing interference, but excessive modification increases stress and shortens life. For spur gears subject to misalignment, higher modification amounts enhance robustness, maintaining fatigue life by alleviating stress concentration. Therefore, designers should balance modification levels based on alignment conditions to optimize the durability and reliability of spur gear systems. Future work could explore dynamic effects and alternative modification curves for spur gears in high-speed applications.