In aviation transmission systems, spiral bevel gears play a critical role in achieving directional changes and power transmission due to their smooth operation, high load capacity, low noise, and compact structure. However, under high-speed and heavy-load conditions, spiral bevel gears are susceptible to traveling wave resonance, which can exacerbate wear on gear contact surfaces and subject the web to cyclic alternating stresses, leading to fatigue damage and potential structural failure. This study focuses on investigating the traveling wave resonance characteristics of a high-speed spiral bevel gear used in aviation applications to identify safe operating speed ranges and enhance system reliability.
The research employs finite element analysis (FEA) to model the driven spiral bevel gear and perform modal analysis. The goal is to determine the natural frequencies and mode shapes, particularly the nodal diameter modes, within the meshing excitation frequency range. By constructing a Campbell diagram based on modal data, resonance points are identified, and resonance margins are calculated to assess the safety of the gear at its rated operating speed. The findings provide insights into avoiding dangerous speed intervals and ensuring operational integrity.
The driven spiral bevel gear is analyzed using a finite element model created in CATIA and simulated in ANSYS Workbench. The gear’s material properties and structural parameters are summarized in Tables 1 and 2. The mesh model, consisting of 810,186 elements with a quality of 0.756, ensures accuracy in resonance analysis. Constraints are applied to simulate bearing supports, and rotational speed conditions are set to mimic operational scenarios, including acceleration to the rated speed of 7,626 r/min.
| Parameter | Driven Spiral Bevel Gear |
|---|---|
| Gear Ratio | 27/74 |
| Module | 3.85 |
| Face Width (mm) | 38.5 |
| Helix Angle (°) | 35 |
| Hand of Spiral | Right-hand |
| Material | Elastic Modulus (GPa) | Poisson’s Ratio | Density (kg/m³) | Tensile Strength (MPa) | Yield Strength (MPa) |
|---|---|---|---|---|---|
| 20CrMnTi | 207 | 0.25 | 7,850 | 1,080 | 885 |
Modal analysis reveals the natural frequencies and mode shapes of the driven spiral bevel gear. The meshing excitation frequency $$f_m$$ is calculated using the formula:
$$f_m = \frac{n_1 z_1}{60},$$
where $$n_1$$ is the rotational speed of the driven spiral bevel gear in r/min, and $$z_1$$ is the number of teeth on the driven gear. For this study, $$z_1 = 74$$, and the external excitation is generated by the driving gear with $$z_2 = 27$$ teeth, resulting in an excitation multiplier of 27 for the Campbell diagram. The first 30 modal orders are analyzed to cover the meshing frequency range up to approximately 8,900 Hz.
The results indicate six distinct nodal diameter modes within the excitation frequency range, each associated with forward and backward traveling waves. Table 3 summarizes the natural frequencies and corresponding mode shapes for the first 30 orders. Notably, each nodal diameter mode corresponds to two closely spaced frequencies, representing the forward and backward traveling waves.
| Order | Frequency (Hz) | Mode Shape | Order | Frequency (Hz) | Mode Shape |
|---|---|---|---|---|---|
| 1 | 110.6 | – | 16 | 5,993.8 | – |
| 2 | 1,115.6 | 1 Nodal Diameter | 17 | 5,994.5 | – |
| 3 | 1,115.6 | 1 Nodal Diameter | 18 | 6,393.9 | 5 Nodal Diameters |
| 4 | 1,261.0 | 2 Nodal Diameters | 19 | 6,394.7 | 5 Nodal Diameters |
| 5 | 1,261.1 | 2 Nodal Diameters | 20 | 6,989.9 | – |
| 6 | 1,470.7 | – | 21 | 6,990.2 | – |
| 7 | 2,573.8 | 3 Nodal Diameters | 22 | 7,064.9 | – |
| 8 | 2,574.0 | 3 Nodal Diameters | 23 | 8,012.9 | – |
| 9 | 3,603.3 | – | 24 | 8,015.0 | – |
| 10 | 3,605.7 | – | 25 | 8,416.8 | 6 Nodal Diameters |
| 11 | 4,405.7 | 4 Nodal Diameters | 26 | 8,417.9 | 6 Nodal Diameters |
| 12 | 4,411.3 | 4 Nodal Diameters | 27 | 8,540.8 | – |
| 13 | 5,661.0 | – | 28 | 8,541.0 | – |
| 14 | 5,661.7 | – | 29 | 8,890.9 | – |
| 15 | 5,709.1 | – | 30 | 8,891.4 | – |
The nodal diameter modes are critical as they can lead to resonance and structural damage. Figure 1 illustrates the total deformation plots for the six nodal diameter modes obtained from the modal analysis. These modes include 1, 2, 3, 4, 5, and 6 nodal diameters, with each mode showing radial nodal lines that indicate potential resonance risks.
To assess resonance risks, a Campbell diagram is plotted for the first 30 orders with an excitation multiplier of 27, as shown in Figure 2. The diagram plots natural frequencies against rotational speed, with lines representing forward and backward traveling waves. Intersections between these waves and the external excitation line indicate potential resonance points. However, only intersections corresponding to nodal diameter modes are considered critical.
The Campbell diagram reveals resonance points at specific speeds for the 3-nodal diameter and 4-nodal diameter modes. The resonance speeds are identified as 5,218.6 r/min (backward traveling wave of 3-nodal diameter), 6,330.9 r/min (forward traveling wave of 3-nodal diameter), and 8,758.6 r/min (backward traveling wave of 4-nodal diameter). The resonance margin $$\delta$$ is calculated using the formula:
$$\delta = \left| \frac{n_i – n_0}{n_0} \right| \times 100\%,$$
where $$n_i$$ is the resonance speed and $$n_0$$ is the rated operating speed (7,626 r/min). For aviation spiral bevel gears, a resonance margin greater than 10% is required to ensure safe operation. Table 4 summarizes the resonance margins for the critical points.
| Order | Mode Shape | Traveling Wave Type | Resonance Speed (r/min) | Rated Speed (r/min) | Resonance Margin (%) |
|---|---|---|---|---|---|
| 7 | 3 Nodal Diameters | Backward | 5,218.6 | 7,626 | 31.57 |
| 8 | 3 Nodal Diameters | Forward | 6,330.9 | 16.98 | |
| 11 | 4 Nodal Diameters | Backward | 8,758.6 | 14.85 |
All resonance margins exceed 10%, confirming that the driven spiral bevel gear operates safely at the rated speed. However, for additional testing within 60% to 130% of the rated speed (4,575.6 to 9,913.8 r/min), dangerous speed ranges must be avoided. Based on the resonance points, the dangerous speed intervals are 4,744.18 to 5,798.44 r/min (3-nodal diameter backward wave), 5,755.36 to 7,034.33 r/min (3-nodal diameter forward wave), and 7,962.36 to 9,731.78 r/min (4-nodal diameter backward wave). Thus, the safe operating speed ranges for testing are 4,575.6 to less than 4,744.18 r/min, greater than 7,034.33 to less than 7,962.36 r/min, and greater than 9,731.78 to 9,913.8 r/min.
In conclusion, this study demonstrates the importance of modal analysis and Campbell diagram construction in evaluating traveling wave resonance for high-speed spiral bevel gears. The identified safe speed ranges ensure reliable operation and prevent resonance-induced failures in aviation applications. Future work could explore damping techniques or structural optimizations to further enhance the performance of spiral bevel gears under extreme conditions.