1. Introduction
Spiral bevel gears are widely used in the aviation and vehicle industries for their excellent performance, such as high 重合度，strong load – carrying capacity, low vibration, and low noise. However, during the transmission process, they may be affected by internal and external excitations, resulting in abnormal vibrations, which may lead to amplitude exceeding the limit and even serious failures. Therefore, it is necessary to study the vibration mechanism of spiral bevel gear systems and analyze their modal characteristics. This paper takes a spiral bevel gearbox of an aero – engine as the research object and proposes a set of methods for gear system – level modal analysis of bevel gears.
Currently, the research on the modal analysis of gear systems mainly focuses on spur gears and helical gears, and there are relatively few literatures on the modal analysis of bevel gears. Although some scholars have carried out relevant research, there are still some limitations, such as only analyzing the modal of a single spiral bevel gear, connecting the meshing of gear pairs in a binding way, or using theoretical simulation tooth surfaces that are quite different from actual tooth surfaces, resulting in a large gap between the calculation results and the actual test situation.
2. Accurate Modeling of Spiral Bevel Gears
2.1 Obtaining Tooth Points
The formation of the tooth surface of bevel gears is the result of the combined action of design parameters and machining parameters. When modeling the tooth surface of bevel gears, accurate tooth point information is required. Tooth points can be obtained through two main methods: simulation from machining parameters or tooth surface sampling using a coordinate measuring machine. In the case of no machining errors considered, the tooth point information obtained by these two methods is consistent.
In this paper, the method of obtaining tooth points by simulating machining parameters is adopted. Based on the gear design parameters (as shown in Table 1) and machining parameters (as shown in Table 2), the parameters are imported into the Masta software. After completing the macro – parameter modeling, the machining parameters in Table 2 are input into the spiral bevel gear manufacturing module. After establishing the accurate tooth surface, 7×9 tooth point information is output. A tooth point set and a tooth root circle point set are established according to the modeling requirements.
| Parameter | Small Wheel | Large Wheel |
|---|---|---|
| Number of Teeth | 35 | 37 |
| Module/mm | – | – |
| Mid – point Helix Angle/(°) | – | – |
| Normal Pressure Angle/(°) | – | – |
| Shaft Intersection Angle/(°) | 80 | – |
| Tooth Width/mm | 24 | – |
| Hand of Helix | Right – hand | Left – hand |
| Outer Cone Distance/mm | 89.634 | – |
| Pitch Cone Angle/() | 38.6648 | 41.3352 |
| Face Cone Angle/(°) | 40.859 | 43.3855 |
| Root Cone Angle/(°) | 36.6145 | 39.141 |
| Tooth Root Height/mm | 3.434 | 3.209 |
| Tooth Tip Height/mm | 2.607 | 2.833 |
Table 1: Design Parameters
| Parameter | Large Wheel | Small Wheel Concave Surface | Small Wheel Convex Surface |
|---|---|---|---|
| Blank Swivel Angle/(°) | 61.7443 | 61.5715 | 60.265 |
| Offset Distance/mm | 0 | 0 | 0 |
| Machine Center to Workpiece Mounting Datum Surface/mm | 0 | 7.81848 | – 5.42884 |
| Blank Mounting Angle/(°) | 39.3737 | 36.62 | 36.62 |
| 2 – order Modification Coefficient | – 0.0129735 | 0.047363 | – 0.067056 |
| 3 – order Modification Coefficient | 0.0179555 | 0.0240835 | 0.0006225 |
| Cutter Head Radius/mm | 70.30805 | 74.78439 | 68.1639 |
| Rolling Ratio | 1.478227 | 1.709616 | 1.507862 |
| Saddle/mm | – 0.21505 | – 4.66347 | 3.23934 |
| Cutter Rotation Angle/(°) | 206.44 | 25.5 | 27 |
| Cutter Tilt Angle/(°) | – 0.0561 | 0 | 0 |
| Inner Cutting Edge Angle/(°) | 22.5016 | – | 21.45 |
| Inner Tool Tip Fillet Radius p/mm | 0.7 | – | 74.676 |
| Outer Cutting Edge Angle/(°) | 20.8422 | 19.5 | – |
| Outer Tool Tip Fillet Radius p/mm | 0.7 | 79.121 | – |
| Cutter Tip Width/mm | 1.36 | – | – |
| Radius R/mm | 76.2 | – | – |
Table 2: Machining Parameters
2.2 3D Modeling of Gear Pairs
After obtaining the tooth point information, it is imported into Catia for 3D finite – element modeling. The specific modeling process includes importing tooth point parameters, establishing spline curves, creating mesh surfaces, establishing tooth blanks, and performing operations such as segmentation and array. Finally, a gear model with an accurate tooth surface is obtained.
Then, a complete gear shaft model is established and assembled to obtain a complete gear pair analysis model. All gear shaft materials are 9310 steel, with an elastic modulus of 204 GPa, a density of 7850 kg/m³, and a Poisson’s ratio of 0.293.
2.3 3D Modeling of Gear Systems
After completing the gear pair modeling, it is necessary to introduce stiffness support on the bearing support surface, as shown in Figure 2. The stiffness of the fixture and the actual support stiffness are difficult to be exactly the same, but the order of magnitude can be ensured to be the same. For comparison with the test, the fixture stiffness is used for calculation during simulation, and the stiffness values are shown in Table 3. The contact surface of the gear pair is set as friction contact, with a friction coefficient .
| Bearing Position | X – direction/(N/mm) | Y – direction/(N/mm) | Z – direction/(N/mm) |
|---|---|---|---|
| B1 | 65600 | 1.662×10³ | 18500 |
| B2 | 82200 | 2.086×10³ | 28400 |
| B3 | 2.759×10³ | 8.11×10³ | 46600 |
| B4 | 83000 | 2.648×10³ | 51300 |
3. Modal Analysis
Modal analysis cannot consider non – linear factors such as friction contact, and the binding constraint method is quite different from the actual tooth coupling form. Therefore, this paper adopts the calculation method of prestressed modal, that is, first performing static strength analysis, and then performing prestressed modal calculation on the gear system.
3.1 Static Strength Analysis
When calculating the static strength, for the convenience of comparison with the test, the convex surface of the small wheel drives the concave surface of the large wheel. A torque of 100 N·m is applied to the driving wheel, and the driven wheel is restricted from rotating. The contact situation of the gear pair is analyzed. The calculated contact impression and stress nephogram of the gear pair are shown in Figure 3 and Figure 4 respectively. It can be seen from the figures that the gear pair is in normal contact, with two pairs of teeth participating in the contact, and the maximum stress value is 275.14 MPa.
3.2 Prestressed Modal
Taking the static strength calculation result as the initial condition, the modal of the system within 22 kHz can be calculated. The selected typical modes are shown in Table 4. It can be seen from the table that different from the modal analysis results of a single gear shaft, the prestressed modal analysis results show that the number of diametral – pitch – type vibration modes increases, and the first – to – fourth – order diametral – pitch – type vibration modes all correspond to multiple modal frequencies. Due to the influence of contact, a special coupled vibration mode such as the diametral – pitch – type coupled conical (order 12) even appears. Some vibration mode diagrams are listed in Section 4.3.
| Order | Frequency/Hz | Vibration Mode | |
|---|---|---|---|
| Driving Wheel | Driven Wheel | ||
| 1 | 4063.9/4314.1 | First – order diametral – pitch | – |
| 2 | 5508.7 | – | First – order diametral – pitch |
| 3 | 5705.1 | Second – order diametral – pitch | First – order diametral – pitch |
| 4 | 6022.8 | Second – order diametral – pitch | – |
| 5 | 6172.5 | Second – order diametral – pitch | Second – order diametral – pitch |
| 6 | 6361.3 | Coupled | Second – order diametral – pitch |
| 7 | 6541.6 | Coupled | Second – order diametral – pitch |
| 8 | 6543.7 | Conical | Second – order diametral – pitch |
| 9 | 6559.5 | Coupled | Second – order diametral – pitch |
| 10 | 7241.3 | Second – order diametral – pitch | Second – order diametral – pitch |
| 11 | 8159.7 | – | Conical |
| 12 | 8778.6 | Coupled | Conical + diametral – pitch |
| 13 | 12048 | Third – order diametral – pitch | – |
| 14 | 12220 | Third – order diametral – pitch | Third – order diametral – pitch |
| 15 | 12650 | Torsional | Third – order diametral – pitch |
| 16 | 12849/12926 | – | Third – order diametral – pitch |
| 17 | 13731 | Third – order diametral – pitch | Third – order diametral – pitch |
| 18 | 17644/17729 | Conical | Coupled |
| 19 | 19360/19458 | Fourth – order diametral – pitch | – |
| 20 | 20853/21082 | – | Fourth – order diametral – pitch |
Table 4: Typical Modes
4. Experimental Verification
4.1 Experimental Setup
The experiment uses a fixture to replace the support of the casing and bearings, and the stiffness value is shown in Table 3. The test piece of the modal experiment is shown in Figure 5. A torque of 100 N·m is applied by using a loading rod to ensure the meshing of the gear pair, and then it is locked with a fixture and an anti – rotation key. After removing the loading rod, the modal experiment is carried out, as shown in Figure 6. During the experiment, the mobile force hammer method based on the LMS software is used for testing, that is, single – point excitation and multi – point response. The process is shown in Figure 7. The layout of the hammering points is shown in Figure 8. The 4th and 7th hammering points of the driving wheel and the 3rd and 24th hammering points of the driven wheel are the sensor pasting positions. During the experiment, for the gear shaft, the bending vibration mode is mainly concerned, and the hammering direction is set as the radial direction; for the gear tooth surface, the diametral – pitch vibration mode is mainly concerned, and the hammering direction is set as the axial direction, and the vibration modes below the fifth – order diametral – pitch are mainly concerned. Considering the test requirements and the operability of the hammering points, 10 hammering points are evenly arranged. The final geometric model .
4.2 Experimental Data Analysis
The extraction of modal parameters adopts the curve – fitting method, and the optimal estimates of natural frequency, damping, and vibration mode are obtained based on the least – square estimation. The modes with stable extraction of frequency, damping, and vibration mode vectors are shown in Table 5. The typical vibration modes are listed in Section 4.3.
The Modal Assurance Criterion (MAC) is used to verify the correlation of the measured modes (Figure 10). It can be seen from Figure 10 that except for the diagonal elements, the MAC values are relatively small, that is, the correctness of the selected modes is high, and the experimental results are reliable.
| Order | Frequency/Hz | Vibration Mode | |
|---|---|---|---|
| Driving Wheel | Driven Wheel | ||
| 1 | 4782 | First – order diametral – pitch + bending | – |
| 2 | 5416 | Second – order diametral – pitch | – |
| 3 | 5760 | Second – order diametral – pitch | Second – order diametral – pitch |
| 4 | 6429 | Second – order diametral – pitch + bending | – |
| 5 | 6633 | Second – order diametral – pitch + bending | Second – order diametral – pitch + bending |
| 6 | 6685 | Second – order diametral – pitch + bending | Second – order diametral – pitch + bending |
| 7 | 7562 | Conical + bending | – |
| 8 | 11878 | Third – order diametral – pitch | – |
| 9 | 12558 | – | Third – order diametral – pitch |
| 10 | 17381 | Bending | – |
| 11 | 19159 | Fourth – order diametral – pitch + bending | – |
| 12 | 20025 | Fourth – order diametral – pitch + bending | – |
4.3 Comparison between Simulation and Experiment
Combining Table 4 and Table 5, the simulation calculation results are compared with the experimental results. It is found that the simulation calculation results and the measured modes do not have a one – to – one correspondence. The reasons are as follows: ① The influence of the fixture during the experiment leads to the appearance of vibration modes related to the fixture or coupled vibration modes; ② Due to human factors or conditions, some modes are not excited.
This paper mainly compares the vibration modes of the gear pair, ignoring the influence of the fixture on the bending vibration mode. The results are shown in Table 6 and Figure 11. It can be seen that except for the first – order diametral – pitch vibration mode of the driving wheel, the simulation calculation results are larger than the measured values. In terms of vibration modes, the measured first – order diametral – pitch vibration mode is obviously coupled with bending, and the correspondence with the simulation results is not strong; the influence of bending on other vibration modes is relatively small, and the errors are all less than 10.08%. Considering that the modal calculation margin in engineering is about 10%, the theoretical calculation results can be used to support the experiment.
| Order | Vibration Mode | Simulation Frequency/Hz | Measured Frequency/Hz | Error/% |
|---|---|---|---|---|
| Driving Wheel | Driven Wheel | |||
| 1 | First – order diametral – pitch | 4314.1 | 47 |
5. Modal Sensitivity Analysis
In engineering, both the convex and concave surfaces of aviation bevel gears are working surfaces, and the load is not constant. Therefore, it is necessary to analyze the influence of the contact surface and load on the mode to determine whether multi – condition analysis is required in actual analysis. Four groups of working conditions shown in Table 7 are analyzed, with group 3 as the reference benchmark.
| Group | Working Surface | Driving Wheel Load/(N·m) |
|---|---|---|
| 1 | Small Wheel Convex Surface – Large Wheel Concave Surface | 100 |
| 2 | Small Wheel Concave Surface – Large Wheel Convex Surface | 50 |
| 3 | Small Wheel Concave Surface – Large Wheel Convex Surface | 100 |
| 4 | Small Wheel Concave Surface – Large Wheel Convex Surface | 130 |
Table 7: Analysis Conditions
5.1 Influence of Different Contact Surfaces on Prestressed Modes
Combining group 1 and group 3, the gear system modes under the condition of only different contact surfaces are analyzed. The vibration modes are basically the same as those in Table 4, and the natural frequency errors are shown in Table 8. It can be seen from Table 8 that after changing the contact surface, the system modes change. Most of the modes have the same vibration form, and the natural frequency error does not exceed 3%, but the vibration forms of some modes change. The vibration mode diagrams are listed in Figure 12, and new vibration modes (order 11) and new – form coupled vibration modes (order 7 and order 19) appear. Therefore, when simulating and predicting the experiment, it is necessary to conduct modal analysis for different contact surfaces respectively.
| Order | Group 1 Calculated Frequency/Hz | Group 3 Calculated Frequency/Hz | Error/% |
|---|---|---|---|
| 1 | 4063.9/4314.1 | 4112.4/4315 | -1.18/-0.02 |
| 2 | 5508.7 | 5500.8 | 0.14 |
| 3 | 5705.1 | 5644.8 | 1.07 |
| 4 | 6022.8 | 6042.7 | -0.33 |
| 5 | 6172.5 | 6164.8 | 0.12 |
| 6 | 6361.3 | 6406.2 | -0.70 |
| 7 – 1 | – | 6536.9 | – |
| 7 – 2 | 6541.6 | – | – |
| 8 | 6543.7 | 6543.2 | 0.01 |
| 9 | 6559.5 | 6563.2 | -0.06 |
| 10 | 7241.3 | 7464.4 | -2.99 |
| 11 | – | 7886 | – |
| 12 | 8159.7 | 8146.7 | 0.16 |
| 13 | 8778.6 | 8983 | -2.28 |
| 14 | 12048 | 12060 | -0.10 |
| 15 | 12220 | 12305 | -0.69 |
| 16 | 12650 | 12649 | 0.01 |
| 17 | 12849/12926 | 12844/12940 | 0.04/-0.11 |
| 18 | 13731 | 13661 | 0.51 |
| 19 – 1 | – | 17621 | – |
| 19 – 2 | 17644/17729 | – | – |
| 20 | 19360/19458 | 19364/19530 | -0.02/-0.37 |
| 21 | 20853/21082 | 20849/21035 | 0.02/0.22 |
5.2 Influence of Different Loads on Prestressed Modes
Combining group 2, group 3, and group 4, the gear system modes under the condition of only different loads are analyzed. The vibration modes are basically the same as those in Table 4, and the errors are shown in Table 9. It can be seen from Table 9 that after changing the load, the change of the system mode is only reflected in the change of frequency, and the vibration form remains the same. The larger the load, the higher the natural frequency under the same vibration form, but the error value does not exceed 2.5%. Therefore, when simulating and predicting the experiment, within a certain load range, the influence of the load on the bevel gear system mode can be ignored.
| Order | Group 2 Calculated Frequency/Hz | Group 3 Calculated Frequency/Hz | Error 1/% | Group 4 Calculated Frequency/Hz | Error 2/% |
|---|---|---|---|---|---|
| 1 | 4104.3/4299.7 | 4112.4/4315 | -0.20/-0.35 | 4122.5/4316.2 | 0.25/0.03 |
| 2 | 5500.2 | 5500.8 | -0.01 | 5501.3 | 0.01 |
| 3 | 5623 | 5644.8 | -0.39 | 5646.2 | 0.02 |
| 4 | 6029.5 | 6042.7 | -0.22 | 6048.4 | 0.09 |
| 5 | 6159.4 | 6164.8 | -0.09 | 6166.7 | 0.03 |
| 6 | 6403.4 | 6406.2 | -0.04 | 6411.3 | 0.08 |
| 7 | 6536.3 | 6536.9 | -0.01 | 6537.1 | 0.00 |
| 8 | 6543 | 6543.2 | 0.00 | 6543.1 | 0.00 |
| 9 | 6549.3 | 6563.2 | -0.21 | 6570.7 | 0.11 |
| 10 | 7359.7 | 7464.4 | -1.40 | 7494.4 | 0.40 |
| 11 | 7878.1 | 7886 | -0.10 | 7889.3 | 0.04 |
| 12 | 8142.9 | 8146.7 | -0.05 | 8147.2 | 0.01 |
| 13 | 8766.6 | 8983 | -2.41 | 9047 | 0.71 |
| 14 | 12052 | 12060 | -0.07 | 12063 | 0.02 |
| 15 | 12774 | 12305 | -0.25 | 12327 | 0.18 |
| 16 | 12647 | 12649 | -0.02 | 12650 | 0.01 |
| 17 | 12842/12929 | 12844/12940 | -0.02/-0.09 | 12846/12941 | 0.02/0.01 |
| 18 | 13395 | 13661 | -1.95 | 13748 | 0.64 |
| 19 | 17619 | 17621 | -0.01 | 17623 | 0.01 |
| 20 | 19360/19511 | 19360/19458 | -0.02/-0.10 | 19366/19538 | 0.01/0.04 |
| 21 | 20848/20971 | 20853/21082 | 0.00/-0.30 | 20850/21050 | 0.00/0.07 |
Table 9: Influence of Load on Modes
6. Conclusion
(1) Taking a spiral bevel gearbox of an aero – engine as the research object, a 3D model of a spiral bevel gear pair with an accurate tooth surface is obtained based on design parameters and machining parameters, and a set of methods for gear system – level modal analysis of bevel gears is proposed.
(2) By comparing the simulated calculation modes with the experimental results, the error between the vibration modes of the gear pair mainly composed of diametral – pitch – type and its coupled vibration modes and the experiment is less than 10.08%, which verifies the accuracy and effectiveness of the calculation method.
(3) The results show that the bevel gear system mode is more sensitive to the contact surface and relatively less sensitive to the load. The reason is that different contact surfaces represent completely different meshing states, and the meshing stiffness values may differ by an order of magnitude. While the change of the load on the meshing state is only reflected in the size of the stressed surface, and the influence on the meshing stiffness does not cross an order of magnitude. Therefore, within a certain range, the influence of the load can be ignored, but modal analysis is required for different contact tooth surfaces respectively.
