In calculating the dynamic meshing force of the gear, the influence of the degree of freedom of swing is neglected, and the gear pair is coupled by the meshing stiffness, meshing damping and tooth side clearance along the meshing line.First, the analytical expression of the dynamic tooth side clearance caused by the change of center distance is derived. Then, considering the changes of tooth profile modification, speed, center distance and meshing state of tooth surface and back, the engagement position and boundary conditions of single and double teeth engagement and profile modification are determined by using the pressure angle of engagement point, and the dynamic meshing stiffness is calculated by using the analytical method. Finally, the dynamic tooth side clearance and tooth profile modification are considered.The dynamic engagement stiffness replaces the time-varying engagement stiffness of the constant clearance and quasi-static period in the previous engagement model, and a non-linear dynamic engagement model of gear is put forward.

In gear meshing drive, to avoid jamming, there must be clearance between the tooth profiles.Side clearance is defined as the minimum distance of the non-contact tooth surface along the meshing line when one side of the tooth surface contacts.Side clearance can be changed by adjusting the length of the common normal or the center distance.From the analysis, it can be seen that geometric eccentricity, center distance deviation and transverse displacement of gears will cause dynamic changes of center distance.At the same time, the change of center distance will cause the dynamic change of tooth side clearance.

The tooth side clearance of the external gear pair is shown in the figure. In the figure, L(t) is the dynamic center distance and alpha(t) is the dynamic engagement angle. They are all functions of time in the process of gear transmission and are calculated by formula.According to the engagement principle of involute gear, the dynamic backlash is: