When calculating the meshing stiffness of single tooth with finite element method, first extract the contact force resultant force of spiral bevel gear tooth surface of intermediate tooth, as shown in Figure 1. Then the comprehensive deformation of each meshing instantaneous is extracted. As shown in Figure 2, the colored elliptical area is the contact area on the surface of a passive gear tooth at the moment of meshing. It can be approximately considered that the contact force acts on the elliptical center with the maximum contact pressure. Therefore, the displacement of the center of the contact ellipse can be extracted as the comprehensive deformation of the meshing instantaneous. Establish two dynamic coordinate systems which are respectively fixedly connected with the active and passive gears, and extract the displacement of the nodes in the contact area of the large and small tooth surfaces of the spiral bevel gear in the dynamic coordinate system, then the single tooth deformation curve of the active and passive teeth of the spiral bevel gear corresponding to the contact force can be obtained. The single tooth deformation of the active and passive gears of the spiral bevel gear can be superimposed to obtain the single tooth comprehensive deformation curve, as shown in Figure 3:
The meshing stiffness value of each meshing position can be calculated by the formula, so as to obtain the single tooth meshing stiffness curve, as shown in Fig. 3.
The influence of coincidence degree should be considered in the comprehensive meshing stiffness of gears. Due to the complexity of point meshing transmission of spiral bevel gear with free-form surface, the theoretical formula of tooth pairs participating in meshing at each instant can not be calculated accurately. It can be calculated indirectly through the following methods: output the contact force curve of two adjacent teeth at the same time, as shown in Figure 1. The time interval between two adjacent teeth entering meshing in turn can be read on the abscissa Δ t ； Then the single tooth meshing stiffness curve is calculated according to the time interval Δ T translation, the meshing stiffness curve of the intermediate tooth will cross the stiffness curve of the two adjacent teeth. The degree of intersection can reflect the number of teeth participating in meshing at the same time at a certain moment. The comprehensive meshing stiffness curve can be obtained by superimposing the single tooth meshing stiffness curve, as shown in Figure 4.
