Boundary conditions and finite element mesh generation of central bevel gear

The physical parameters of the material used for the central bevel gear at working temperature are: elastic modulus, Poisson’s ratio and material density. Solid186 element with nodes in hexahedron in ANSYS software is used to mesh the bevel gear structure. In order to calculate the meshing stiffness of gear teeth, when meshing, the meshing of bevel gear teeth is dense to ensure the corresponding calculation accuracy. The finite element mesh is divided by ansa software, and the finite element model of driving bevel gear is obtained, as shown in Figure 1 (a), with a total of 75000 elements; The finite element model of driven bevel gear is shown in Figure 1 (b), with a total of 69000 elements.

(a) Driving bevel gear (b) Driven bevel gear

Although the constraint position of the natural frequency calculation of bevel gear is determined, the selection of nodes on the constraint surface affects the calculation results of the natural characteristics of bevel gear. In order to explore the influence of boundary constraints on the calculation results of natural characteristics of aeroengine bevel gear, different constraint boundary conditions will be used to calculate the natural frequency of aeroengine central bevel gear. The boundary conditions suitable for the calculation and analysis of natural frequency of aeroengine central bevel gear are found. In the finite element model, there are 7 cycles of nodes in plane a, 3 cycles of nodes in plane B, 9 cycles of nodes in plane C and 14 cycles of nodes in plane D. The natural frequency of bevel gear is calculated by using the six constraint boundary method. The number of axial constraint nodes is kept unchanged in the six boundary conditions, and the number of node turns on surfaces a, C and D is reduced successively until 0. The variation curve of the first three order vibration frequency values of each pitch diameter with the constraint type is shown in figures 2 to 6.

From the variation diagram of the natural frequency of the bevel gear with the constraint mode, it can be seen that the higher the pitch diameter of the central bevel gear of the aeroengine, the less sensitive the natural frequency calculation result to the constraint boundary conditions. At the same time, the higher the vibration order, the natural frequency calculation results are also insensitive to the constrained boundary conditions. For example, the calculation results of the first three natural frequencies of 2-pitch diameter, 3-pitch diameter and 4-pitch diameter are basically the same under the six boundary conditions. For vibration with 0 pitch diameter, the first-order natural frequency is greatly affected by boundary constraints. Under the condition of only axial constraints, the first-order natural frequency has been zero. The second-order natural frequency of a pitch diameter changes greatly with the boundary conditions. The first-order vibration modes of 0-pitch diameter and the second-order vibration modes of 1-pitch diameter are shown in Figure 7:

According to the vibration mode diagram (animation display judgment) of the first-order vibration mode with 0 pitch diameter, the vibration mode of this form is the torsional vibration mode of bevel gear, so when the circumferential constraint is removed from the boundary constraint conditions, the vibration frequency of this order will be reduced to zero. From the second-order vibration mode of 1 node diameter, it can be seen that the constraints of plane a and plane D inhibit the vibration mode. When the constraints are weakened, the vibration frequency will be reduced.

The dynamic frequency and static frequency of the central bevel gear of an aeroengine are calculated by using the No. I boundary constraint condition. In the dynamic frequency calculation process, the speed of the bevel gear is considered, and the meshing force of the gear is distributed to each tooth in a draw to calculate the dynamic frequency of the bevel gear.

It can be seen that there is little difference between the dynamic frequency and static frequency calculation results of aeroengine central bevel gear. Therefore, the static frequency calculation value can be used to replace the dynamic frequency value in the process of aeroengine central bevel gear inherent characteristics and traveling wave resonance analysis, so as to reduce the calculation workload.

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