Based on the energy method, the relationship between the z-contact stiffness, bending stiffness, shear stiffness, radial compression deformation stiffness and the stiffness corresponding to the flexible deformation of the gear matrix and the angle of the driving wheel is derived based on the energy method. The change law of the single tooth meshing area and the double tooth meshing area is analyzed, and the change law of the time-varying meshing stiffness with the driving wheel angle is obtained Based on the calculation method of time-varying meshing stiffness of normal gear pair, the tooth root crack fault is introduced to form the time-varying meshing stiffness calculation method of gear pair with root crack fault, and the influence of root crack on the time-varying meshing stiffness of gear pair is compared and analyzed; 8 In this paper, the dynamic model of the involute spur gear pair with degree of freedom is studied. The conclusion is as follows

1) In the process of gear pair meshing, the number of teeth participating in meshing is periodic, which results in the time-varying meshing stiffness of gear pair divided into single tooth meshing area and double tooth meshing area. The meshing stiffness of single tooth meshing area is obviously less than that of double tooth meshing area, and the meshing stiffness at the junction of single tooth meshing area and double tooth meshing area has obvious mutation phenomenon;

2) The results show that the root crack can reduce the time-varying meshing stiffness of the gear pair. With the meshing point gradually moving from the tooth root to the top of the tooth, the influence of the crack depth and angle on the meshing stiffness of the gear pair increases gradually;

3) The root crack has a great impact on the vibration characteristics of the gear pair. The periodic impact wave appears on the time domain diagram, and the side band appears between the meshing frequency and its multiple frequency. With the increase of the crack depth, the impact wave on the time domain diagram will be more obvious, and the side band amplitude on the frequency spectrum diagram will also increase.