The root crack will directly change the effective cross-sectional area and moment of inertia of the simplified cantilever model of gear teeth, which will affect the time-varying meshing stiffness of the gear pair. Therefore, when the energy method is used to solve the time-varying meshing stiffness of the gear pair with root crack, the propagation path of the crack should be determined first, so as to calculate the effective cross-sectional area and moment of inertia of the cantilever beam with crack. Therefore, assuming that the root crack is a full tooth width through crack, a straight line is used to connect the initial position of the root crack with the crack tip, which is simplified as the crack growth path, and the angle between the crack growth path and the symmetry axis of the tooth is 𝜈, as shown in Fig. 1.
The root crack will not change the tooth profile curve, and the effective tooth surface contact line 𝐿 will not change. The tooth can bear the radial load as the normal tooth, and the Hertz contact stiffness and radial compression stiffness will not change, but the bending stiffness and shear stiffness will not remain unchanged. Therefore, the moment of inertia of the tooth section and the effective area of the cantilever beam can be described as follows:
Where, ℎ C is the vertical distance from the crack tip to the center line of the gear tooth; 𝑔 C is the horizontal distance from the crack tip to the tooth profile curve. The expression of ℎ C is as follows:
Referring to the method for calculating the time-varying meshing stiffness of normal gears, as shown in Fig. 2, the influence of root crack on stiffness can be divided into two cases according to the depth of crack, and the bending stiffness and shear stiffness of gear teeth with root crack fault can be discussed in two cases.