# Calculation results of meshing stiffness of helical gears without considering tooth profile trimming

In order to verify the accuracy and rapidity of the method in this paper, the same gear parameters are used for calculation, as shown in the Table 1.

The load per unit tooth width of the gear is 300 n / mm, and the helix angle β Take 5 °, 10 °, 15 °, 20 ° and 25 ° respectively. The calculation results of the mean value Ka of gear meshing stiffness obtained by this method and the other two methods are shown in Table 2. Among them, the finite element method is adopted, and the HB method is calculated according to the standard Hb / z84.1-1984 of the Ministry of aviation industry.

It can be seen from the results in Table 2 that the error between the meshing stiffness calculated by the method and the results calculated by the other two methods is within 8%. The time required to solve the helical gear meshing stiffness by the finite element method is about 120 s, while the calculation time by the method in this paper is only about 10 s, the calculation efficiency is 12 times that of the finite element method, and is not affected by the driving wheel angle division n. As shown in Figure 1, the helical gears with and without axial deformation at different helix angles are given β Average meshing stiffness Ka. When β ＜ At 15 °, the difference between the average meshing stiffness Ka of helical gears with and without axial deformation shall not exceed 10%. With the increase of helix angle, the average meshing stiffness of helical gear without axial deformation is larger and larger than that of helical gear with axial deformation. The difference between the two is more than 10%. Therefore, the traditional material mechanics method to solve the meshing stiffness of helical gear is only suitable for the range of small helix angle. When the helix angle is large, the influence of the axial deformation of helical gear on the meshing stiffness of helical gear must be considered.

Limited to space, only the comprehensive meshing stiffness K is given when the helix angle is 5 °, 15 ° and 25 ° γ The relationship with the total length L of the meshing line is shown in Figure 2. In Fig. 2, T represents the dimensionless time, and T represents the ratio of the time experienced by a tooth from the beginning of meshing to the corresponding position to the meshing cycle.

It can be seen from Figure 2 that the change trend of the meshing line length of helical gear wheel is roughly the same as that of the comprehensive meshing stiffness β When it is 15 °, the meshing stiffness in the multi tooth meshing area of helical gear is greater than that in the few tooth meshing area, and the meshing stiffness in the double tooth meshing area of spur gear is greater than that in the single tooth meshing area. However, the helical gear will be different from the spur gear, as shown in Fig. 2 (a) and Fig. 2 (b) β When it is 5 ° and 25 °, the meshing stiffness of the multi tooth meshing area of the helical gear is less than that of the few tooth meshing area, which is the same as the conclusion.

Comparing figures 2 (a), 2 (b) and 2 (c), it can be seen that only when the helix angle β When it is 25 °, the change trend of meshing line length of helical gear is consistent with that of comprehensive meshing stiffness. The relationship between the meshing stiffness KS of helical gear single tooth pair and the meshing line length L of single tooth pair is given, as shown in Fig. 3.

As can be seen from Figure 3, with the spiral angle β With the increase of helix angle, the change trend of meshing stiffness of single tooth pair helical gear is closer and closer to the change trend of meshing line length of single tooth pair, so the change trend of comprehensive meshing stiffness of helical gear is closer and closer to the change trend of meshing line length with the increase of helix angle. According to the improved algorithm of the position and length of the helical gear meshing line, combined with Fig. 4, it can be seen that the helix angle β = 5 °, 15 °, yes（ ω B2 － ω B1) > b / P, corresponding to Fig. 4 (b), when the helical gear meshing line is between a’c and c’b, the length of the meshing line is equal everywhere, while the meshing stiffness of the single tooth pair of the helical gear is not equal everywhere. Therefore, in helical gear dynamics, it is not accurate to use the change of the meshing line length instead of the change of the meshing stiffness at this time, and the error is large. When helix angle β = At 25 °, yes（ ω B2 － ω B1) ＜ B / P, corresponding to Fig. 4 (a), when the helical gear meshing line is between BC and a’c ‘, the meshing line length and meshing stiffness of the single tooth pair of the helical gear are equal everywhere. At this time, it is easier to replace the change of meshing stiffness with the change of helical gear meshing length（ ω B2 － ω B1) ＜ B / P is more accurate and the error is small.

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