1. Meshing parameters of face gear

The meshing parameters of face gear are: modulus M = 5, pressure angle  = 20 °, number of face gear teeth N2 = 40, number of cutter teeth ns = 20. In order to realize point contact of face gear, the number of teeth of cylindrical gear should be 1-3 less than that of cutter when machining face gear [1-4]. Therefore, in this paper, the number of teeth N1 of cylindrical gear is 17, 18 and 19 respectively, and the contact area of tooth surface is simulated when the nominal tangential force ft = 1kn is applied.

2. Simulation of tooth contact path

Matlab software is used to make corresponding programs to simulate the tooth surface contact of the cylindrical gears with 17, 18 and 19 teeth and 40 teeth respectively. Without considering the influence of manufacturing and installation errors, the contact points on the left and right tooth profiles of the transmission middle face gears are mutually symmetrical. Therefore, the contact point equation is used, The contact point position on the left tooth profile of the transmission middle gear is simulated. The contact trace R2 ( 1) represented by the contact trace equation (5) of face gear engagement is a spatial curve with  1 as a parameter, as shown in Figure 3. It can be seen from Fig. 3 that the contact path of the tooth surface of the cylindrical gear with different tooth numbers and the face gear has almost no change in the position of the tooth surface.

3. Tooth contact simulation

Seven meshing points are selected from the contact trace line with 17 teeth of the cylindrical gear to analyze the actual contact ellipse field, and the results are shown in Figure 4. Through the analysis of the geometric relationship of the contact object surface, it can be seen that the points with the same distance between the object surfaces will form an area similar to an ellipse on the common section [9]. In Figure 4, the contact area of the tooth surface is centered on the meshing point. Because the short axis of the ellipse is very different, the contact region is approximately a line segment. The direction of the contact area from the top to the bottom of the tooth is a group of line segments, which rotate anticlockwise on the tooth length section. It reaches the maximum at the high and middle part of the teeth and is smaller at the top of the teeth. It can be seen from Figure 4 that the face gear belongs to edge contact, which is unfavorable for the face gear transmission and should be avoided as far as possible.

(a) The contact trace of the tooth surface when the cylindrical gear with 17 teeth is meshed with the face gear with 40 teeth;

(b) the contact trace of the tooth surface when the cylindrical gear with 18 teeth is meshed with the face gear with 40 teeth;

(c) the contact trace of the tooth surface when the cylindrical gear with 19 teeth is meshed with the face gear with 40 teeth.

4. Simulation results

Change the number of teeth of face gear to 100, and simulate the contact path of face gear when other parameters are constant. The result is shown in Figure 5. It can be seen from Fig. 5 that changing the transmission ratio of face gear can affect the contact path of tooth face, and the larger the transmission ratio is, the closer the contact path is to the middle of tooth face, the more favorable the face gear meshing is. In other words, large transmission ratio is conducive to improving the transmission quality of face gear, or face gear transmission is suitable for occasions with large transmission ratio.