Research on vibration frequency of double connected gears in aero-engine field

The finite element simulation method is used to calculate and analyze the vibration frequency of aero-engine double connected gears with different common aero-engine gear materials, different numbers of connecting pins, and different numbers of weight-reducing holes. The analysis results are shown that the first several vibration modes of the double connected gears are first caused by the big gear, the vibration frequency of the double connected gears is almost not affected by the number of connecting pins, the vibration frequency of the double connected gears is little affected by different common aero-engine gear materials, while the vibration frequency of double connected gears is reduced with the increase of the number of weight reduction holes. The design and vibration avoidance reference of double connected gears are provided in the paper.


Introduction
The gear transmission system has the requirements of high speed and high reliability, which is the key of aero-engine power transmission and one of the important component systems of aero-engine.In the process of high-speed rotation of the gear, the gear is prone to vibration due to the effect of internal and external excitation and continuous impact.When the gear vibrates, not only the noise is generated, the reliability of the gear is affected, but also the failure of the whole gear transmission system is caused.Therefore, analyzing the vibration characteristics of the transmission gear and obtaining its vibration frequency and mode diagram has important design reference significance for avoiding the vibration point and improving the working reliability of the gear during the structure design.
At present, many scholars have studied the vibration characteristics of gear.Wang Yanfeng et al. [1] studied the vibration characteristics of aero-engine bevel gears and verified them through tests.The results shown that the bevel gears had dangerous resonance in the case of three-segment forward wave and four-segment rear traveling wave.You Yu et al. [2] studied the vibration characteristics of spiral bevel gears under different transmission errors, and the results shown that the vibration noise of bevel gears decreased when the transmission errors were reduced.Luan Xiaochi et al. [3] studied the traveling wave resonance characteristics and its influence rule of spiral bevel gear under parameter adjustment, and the research results shown that adjusting the spoke plate thickness could avoid the resonance speed or frequency.Xue Yuanshui [4] studied the influence of lubricating oil viscosity on gear vibration and came to the conclusion that the thicker the lubricating oil film, the smaller the gear amplitude.Yi Yuanyuan et al. [5] studied the vibration response of gear pairs caused by eccentricity error, and found that meshing related parameters such as center distance, meshing Angle, tooth side clearance and transmission ratio of gear pairs all had an impact on gear vibration.Zou Peng [6] studied the vibration and modal analysis of aviation gear box, and compared the test identification results with the finite element modal analysis results to verify the correctness of the finite element model of aviation gear box and so on.It can be seen that there are relatively few researches on vibration characteristics of aero-engine double connected gears at present.Therefore, based on the finite element simulation method, the influence of gear material, number of connection pins and number of weight reduction holes on vibration characteristics of double connected gears are analyzed and studied, and the guiding basis for the design and vibration avoidance of double connected gears are provided in the paper.

The model of the double connected gears
The double connected gears are composed of gear 1, gear 2 and 6 pins, wherein gear 2 is installed on gear 1 through the pin, and the fit between the pin and gear is interference fit.The double connected gears parameters are shown in Table 1.The deep groove ball bearing is installed on the journal at both ends of gear 1, and the inner surface of one shaft neck is spline connected, which is shown in Figure 1.The three-dimensional model of the double connected gears is shown in Figure 2.

Material properties
At present, gear materials are mainly structural steel in the field of aero-engine, such as 12Cr2Ni4A, 14CrMnSiNi2MoA, 16Cr3NiWMoVNbE and 18Cr2Ni4WA.The six connection pins connecting gear 1 and gear 2 are made of 05Cr17Ni4Cu4Nb.The material properties of these five materials [7] are shown in Table 2.

Dividing the grid
Tetrahedral meshing method is used to divide meshes in Workbench Mechanical module.In order to improve the efficiency of meshing, a simplified treatment was carried out for the double connected gears, and the fillet and chamfering of some non-critical parts are cancelled.Meanwhile, spline joint is replaced by spline dividing circular surface.The divided grid has a total of 133,970 units, and the meshing model is shown in Figure 3.

Loads and boundary conditions
Gear modal analysis is carried out to obtain each order of gear natural frequency and mode shape, without considering the influence of external load on its natural frequency.Therefore, no load is applied.
Deep groove ball bearing is installed at both ends of the journal of the double connected gears, splines are designed on the inner surface of one end of the Cylindrical neck.Radial force is mainly borne by the deep groove ball bearing and circumferential motion is mainly constrained by the spline connection.Therefore, in order to simulate the actual assembly state of double connected gears, radial freedom is constrained in cylindrical support on the fitting surface of the gear and bearing, and circumferential freedom are constrained on the spline indexing circle.
Considering that gear 2 is installed on gear 1 through the pin, the fit between the pin, gear 1 and gear 2 is interference fit.Therefore, the contact mode between the pin, gear 1 and gear 2 is set as friction contact.According to the mechanical design manual, the dry friction coefficient of the two kinds of steel is 0.6~0.7,so the friction coefficient is set as 0.65.

Modal analysis results
In order to compare the influence of different material properties on the natural frequency of double connected gears, four different gear materials in Table 2 are used for modal analysis (except for material properties, other Settings are consistent), and the first several orders of natural frequencies and vibration modes of double connected gears of different materials are obtained, which is shown in Table 3. Note: The (large) after the natural frequency value refers to the big gear, namely gear 2.
Taking 16Cr3NiWMoVNbE material as an example, the vibration modes corresponding to each natural frequency are shown in Figure 4.According to the vibration mode diagram of the double connected gears (16Cr3NiWMoVNbE), the first several vibration modes are generated by gear 2. The vibration energy required by gear 2 (big gear) is smaller than that of gear 1, so the vibration modes of gear 2 (big gear) are preferentially generated.According to Table 3, the natural frequency variation trend diagram of double connected gears with different materials was drawn, which is shown in Figure 5.As can be seen from Table 3 and Figure 5, the corresponding frequencies of each vibration mode of the double connected gears with different materials have small difference.Among them, the frequency difference of one-pitch diameter ranges from 6Hz to 44Hz, the frequency difference of two-pitch diameter ranges from 5Hz to 51Hz, the frequency difference of one pitch circle ranges from 10Hz to 43Hz, the frequency difference of three-pitch diameter ranges from 16Hz to 99Hz, and the frequency difference of four-pitch diameter ranges from 29Hz to 144Hz.
It can be seen that the maximum difference and minimum difference of the natural frequencies of the first several stages of pitch diameters and pitch circle modes of double connected gears of different materials are 144Hz and 5Hz.The material properties of common gear materials for aero-engine have a certain influence on the natural frequencies, but the influence is small.

Vibration characteristics analysis of double connected gears with different number of connection pins
In order to compare the influence of different number of connection pins on the natural frequency of double connected gears, the double connected gears model (6 connection pins) is modified, and the mounting holes of the connection pins are changed to 4 (uniform) and 7 (uniform) respectively, and the corresponding number of connection pins are assembled on the model, the rest of the structure unchanged.The gear material is 16Cr3NiWMoVNbE, and the connection pin material is 05Cr17Ni4Cu4Nb.The material characteristics are shown in Table 2.The mesh division is consistent with the mesh division method adopted in section 1.3, and the boundary conditions are consistent with those in section 1.4.The modal analysis results are shown in Table 4, and the variation trend of the natural frequency of double connected gears with different number of connection pins is shown in Figure 6.As can be seen from Table 4 and Figure 6, the broken line diagram of the natural frequency of double connected gears with different number of connection pins almost coincides, namely, the corresponding natural frequency values of each vibration mode of double connected gears with different number of connection pins are almost consistent.It can be seen that the change of the number of connection pins has almost no influence on the corresponding natural frequency of each vibration mode of double connected gears.

Analysis of vibration characteristics of double connected gears with different number of weight reducing holes
As can be seen from Table 3 and Table 4, the first several vibration modes of the double connected gears are caused by gear 2. Therefore, in order to compare the influence of weight reduction holes on the natural frequency of the double connected gears, 4, 5 and 6 weight reduction holes with diameters of 12mm are created on the model plate of the gear 2, and the rest structures remain unchanged.The material of double gear is 16Cr3NiWMoVNbE, and the connection pin is still 05Cr17Ni4Cu4Nb.The material characteristics are shown in Table 2.The mesh division is consistent with the mesh division method adopted in section 1.3, and the boundary conditions are consistent with those in section 1.4.The modal analysis results are shown in Table 5, and the variation trend of the natural frequency of double connected gears with different number of connection pins is shown in Figure 7.As can be seen from Table 5 and Figure 7, the corresponding natural frequency value of each vibration mode of double connected gears is the largest when there is no weight reducing hole, and the corresponding natural frequency value of each vibration mode of double gear decreases with the increase of the number of weight reducing holes.It can be seen that the change of the number of weight reducing holes has a certain influence on the corresponding natural frequency of each vibration mode of double gear, namely, the corresponding natural frequency value of each vibration mode of double connected gears decreases with the increase of the number of weight reducing holes.

Conclusions
The vibration characteristics of double connected gears with different materials, different number of connection pins and different number of weight loss holes are analyzed and studied in the paper.The research results are shown that: (1) The low-order vibration modes of double connected gears are mainly caused by gear 2 (big gear), whose main vibration modes are pitch diameter and pitch circle.
(2) The change of the number of connecting pins has almost no effect on the corresponding natural frequency of each vibration mode of the double connected gears; The material properties of common gear materials (12Cr2Ni4A, 14CrMnSiNi2MoA, 16Cr3NiWMoVNbE and 18Cr2Ni4WA) have a certain effect on the natural frequency, but the effect is small.
(3) The change of the number of weight reduction holes has a certain influence on the corresponding natural frequency of each vibration mode of double connected gears, and the change law is that the corresponding natural frequency of each vibration mode of double connected gears decreases with the increase of the number of weight loss holes.In the process of double connected gears design, the weight reduction holes on the double connected gears amplitude plate can be increased or reduced according to the design requirements, so as to achieve the purpose of structural improvement and vibration avoidance.

Figure 1 .
Figure 1.Structure diagram of the double connected gears.

Figure 2 .
Figure 2. The three-dimensional model of the double connected gears.

Figure 3 .
Figure 3.The grid model of the double connected gears.

Figure 4 .
Figure 4. Vibration mode diagram corresponding to each order of natural frequency.

Figure 5 .
Figure 5. Natural frequencies of double connected gears of different materials.

Table 4 .
Natural frequencies of double connected gears with different number of connection pins.(large) after the natural frequency value refers to the big gear, namely gear 2.

Figure 6 .
Figure 6.Natural frequency of double connected gears with different number of connection pins.

Table 1 .
The parameters table of double connected gears.

Table 3 .
Natural frequencies of double connected gears of different materials.

Table 5 .
Natural frequency of double connected gears with different number of weight reduction holes.
Note: The (large) after the natural frequency value refers to the big gear, namely gear 2. Figure 7. Natural frequency of double connected gears with different number of weight reduction holes.