The mixed lubrication model is composed of basic equations, Reynolds equation is composed of oil film equation, density equation, viscosity equation and load balance equation, which is solved by composite iterative method, and the local elastic deformation integral is solved by fast Fourier transform method. Elastic modulus of gear material e = 219.78 GPA, viscosity of lubricating oil η 0 = 0.09 Pa.s, viscosity pressure coefficient α= 12.5 GPa–1。 The surface roughness of spiral bevel gear is 0.2 μ m. Root mean square roughness σ= zero point two eight μ m。

The spiral bevel gear used in a helicopter gearbox is used as the research object. The gear blank parameters and machining parameters are listed in Table 1 and table 2 The input speed of the small wheel is 2933 R / min and the torque of the small wheel is 190 n · M. the load distribution calculation of spiral bevel gear adopts the method proposed in document, and its load distribution and maximum Hertz contact pressure are shown in Figure 1 It can be seen from Figure 1 that the maximum Hertz contact pressure decreases gradually from meshing in to meshing out (small wheel angle), and the maximum Hertz contact pressure is as high as 1.8 GPa According to the contact analysis model of spiral bevel gear, the relative sliding speed and coiling speed vector between the tooth surfaces of spiral bevel gear can be obtained by formula, and the results are shown in Fig. 2 It can be seen from Fig. 2 (a) that the entrainment speed increases gradually from meshing in to meshing out, and the entrainment angle decreases gradually The relative sliding velocity first decreases and then increases, the relative sliding velocity of the node is the smallest, and the angle direction of the sliding velocity changes suddenly near the node Through the contact analysis of spiral bevel gear, the contact load, velocity vector and tooth surface contact geometry are obtained. These parameters are input into the hybrid lubrication model, the lubrication analysis of spiral bevel gear can be carried out, and then the friction characteristics of meshing pair under lubrication can be obtained.

The mixed lubrication model of spiral bevel gear comprehensively considers the real contact situation of spiral bevel gear. Compared with the simplified algorithm and empirical formula, the solution time of friction coefficient is longer In order to compare the prediction of friction coefficient by empirical formula and accurate algorithm, Fig. 3 shows the calculation results in this paper and the calculation results of friction coefficient by Benedict and Kelley, Xu and Kahraman It can be seen from Fig. 3 that the friction coefficient of a pair of meshing pairs increases first and then decreases by using the mixed lubrication analysis model from meshing in to meshing out, and reaches the maximum at the node position It can be seen from Fig. 2 (b) that the relative sliding speed first decreases and then increases, while the change trend of friction coefficient is opposite When the contact lubrication zone of spiral bevel gear shows the characteristics of non-Newtonian fluid, the changes of contact load, relative sliding speed, entrainment angle, contact geometry (contact ellipse, radius of curvature, etc.) and oil film thickness will affect the shear rate and ultimate shear stress between tooth surfaces, so as to affect the friction characteristics between tooth surfaces The friction coefficient between the tooth surfaces of spiral bevel gears is affected by many complex parameters. Therefore, it is difficult to comprehensively consider the influence of various parameters when using empirical formula to predict the friction coefficient of spiral bevel gears in the whole meshing cycle.

The friction coefficient obtained by Benedict and Kelley, Xu and Kahraman is relatively small, and the friction coefficient results are quite different near the node This is because in Benedict and Kelley’s empirical formula, the relative sliding speed appears on the denominator, while the relative sliding speed near the node is small, resulting in a sudden increase in the friction coefficient at the node If the relative sliding speed is 0, the friction coefficient obtained by Benedict and Kelley’s empirical formula is infinite, which is not in line with the actual situation The friction coefficient obtained by Xu and Kahraman suddenly decreases near the node because Xu and Kahraman adopt the line contact model, that is, it is considered that the direction of entrainment velocity is consistent with the main direction of contact ellipse (short axis of contact ellipse) Zhang et al. Studied the friction characteristics of spiral bevel gear based on the point contact model in which the coiling speed coincides with the short axis of the contact ellipse. The prediction results of the friction coefficient are shown in Fig. 4 (a), which is similar to the prediction results obtained by Xu and Kahraman using the simplified algorithm, that is, there is a sudden change in the friction coefficient near the node Mohammadpour et al. Established a point contact lubrication model considering the included angle of entrainment speed, and analyzed the change of friction coefficient of spiral bevel gear meshing pair in one meshing cycle. The change of friction coefficient is shown in Fig. 4 (b) The model can simulate the contact of spiral bevel gear relatively truly, but the entrainment angle is assumed to be constant It can be seen from Fig. 4 (b) that the variation trend of friction coefficient of mohammadpour et al. Is similar to the results in this paper It can be seen from Fig. 3 that the empirical formula and simplified algorithm of friction coefficient have poor prediction accuracy near the node of spiral bevel gear, while the solution results of friction coefficient at other meshing positions are similar to the calculation results in this paper.

In order to study the influence of rotating speed on the friction coefficient of spiral bevel gear, the variation of friction coefficient is analyzed when the rotating speed is 0.001 ~ 3000 R / min Fig. 5 shows the effect of rotating speed on friction coefficient and contact load ratio at the meshing position with large wheel angle of 0.0 rad (meshing point) and 0.082 rad In the mixed lubrication state, the contact force of tooth surface is borne by the rough peak and lubricating oil, and the contact load ratio is defined as the ratio of the load borne by the rough peak to the total load It can be seen from Figure 5 that the friction coefficient and load ratio gradually decrease with the increase of rotating speed Zhu et al defined the contact lubrication state through the contact load ratio, that is, when the contact ratio is greater than 85%, it is boundary lubrication or dry contact, when the contact ratio is 0, it is full film lubrication, and when the contact ratio is between the two, it is mixed lubrication According to the change of load ratio, when the speed is less than 0.1 R / min, the meshing pair of spiral bevel gear is in boundary lubrication or dry contact state, and its friction coefficient is about 0.13. When the speed is greater than 1000 R / min, its friction coefficient is about 0.05, and it is in full film lubrication state. Its change trend is consistent with the change law of Stribeck curve obtained by Zhu et al In addition, when the rotating speed is less than 0.1 R / min, the friction coefficient calculation formulas of Benedict and Kelley, Xu and Kahraman have large errors, and their friction coefficient is greater than 0.13 Especially at very low speed, i.e. 0.001 R / min, the friction coefficient is greater than 0.2, which is inconsistent with the numerical range of boundary friction coefficient (0.1 ~ 0.15) This is because the empirical formulas of Benedict and Kelley, Xu and Kahraman are obtained by multi parameter fitting, and the fitting speed is not low enough, resulting in large prediction error of friction coefficient at low speed When the rotating speed is greater than 0.1 M / min, that is, in the state of mixed lubrication and full film lubrication, the calculation results obtained by the simplified algorithm are close to the results.

Fig. 6 shows the change of meshing efficiency of a pair of teeth in a meshing cycle. It can be seen from Fig. 6 that the lowest meshing efficiency at the meshing point is about 96.8% With the increase of large wheel angle, the relative sliding speed decreases and the friction loss decreases, resulting in the increase of meshing efficiency. At the node, the transmission efficiency is the largest, about 98.6%; The rotation angle of the large wheel continues to increase, and the transmission efficiency begins to decrease. Near the meshing point, the transmission efficiency begins to increase again. This is because the coincidence degree of the spiral bevel gear is greater than 1, and the next pair of teeth enter the meshing transmission It is known from Fig. 3 that the error of solving the friction coefficient by using the method near the node is relatively large. However, at the node position, different friction coefficient calculation methods have little effect on the transmission efficiency. This is because the relative sliding speed at the node position is small, and the friction coefficient has little effect on the friction power loss Fig. 7 shows the effect of rotating speed on meshing efficiency when the large wheel angle is 0 and 0.082 rad With the increase of rotating speed, the transmission efficiency increases, which is consistent with the research results At different speeds, similar to the solution of friction coefficient, the meshing efficiency obtained by Xu and Kahraman’s friction coefficient calculation method is close to the calculation results in this paper under mixed lubrication and full film lubrication, while the error of Benedict and Kelley’s empirical formula is relatively large.