## 1 Introduction  
The planetary gear transmission system has the advantages of high transmission efficiency and compactness performance, etc. Therefore, it is widely used in aeronautics and astronautics, wind turbines, construction machinery, and so on; furthermore, it is a crucial component for transferring torque and motion. Due to the change of the pairs and position of gear teeth during the meshing, the mesh stiffness of gear changes constantly. The time-varying characteristic of mesh stiffness is a crucial parameter excitation source, which has a significant influence on the dynamic response characteristic of the gear transmission system. Therefore, how to accurately calculate the tooth mesh stiffness has become a focus of gear dynamics research. So far, scholars have carried out some research on the calculation methods of mesh stiffness in healthy gear. In the early 1940s, [PERSON] and [PERSON] [1] calculated the displacement of a cylindrical gear with load, considering contact deformation, bending deformation, and elastic deformation of tooth transition. [PERSON] [2] proposed a new model based on [PERSON]'s model, taking into account the influence of cylindrical gear flexibility and transitional stress. Later, [PERSON] [3] developed the potential energy method for calculating the total mesh stiffness of spur cylindrical gears. Even though [PERSON] and [PERSON] had considered axial compression deformation, bending deformation, and contact deformation. While they ignored the influence of shear of the gear tooth. Based on this, [PERSON] [4] considered the influence of gear tooth shear deformation, based on [PERSON]. The gear body deformation stiffness has been considered by [PERSON] et al. [5], and the authors had simplified the gear tooth into a cantilever beam on the base circle, and solved and analyzed the mesh stiffness. However, the gear tooth starts from the root circle rather than the base circle, so these previous models ignore the transition between the root circle and the base circle. Based on [PERSON], [PERSON] et al. [6; 7] simplified the gear teeth into the cantilever beam on the base circle and the root circle, respectively, and deduced the solution of the mesh stiffness of planetary gears. [PERSON] et al. [8] took the potential energy of gear tooth transition part between the tooth root circle and the base circle into consideration, and put forward the calculation method of time-varying mesh stiffness (TVMS) with different tooth numbers, and took into account the transmission error of tooth shape, TVMS and bearing stiffness. Based on [PERSON] and [PERSON]'s model, [PERSON] et al. [9] considered the accurate transition curve between the tooth profile and root circle, so as to calculate the TVMS of the gear pair. [PERSON] et al. [10] established an analytical calculation model for the mesh stiffness of the internal gear pair considering the tooth profile displacement. Gear transmission is the most common transmission mode in mechanical equipment. With the increasing requirements of modern machinery for gear transmission, such as high speed, heavy load, special medium, etc., the gear teeth are also more prone to failure. There are many forms of gear failure, among which gear tooth cracks are the most common failure forms. Once the gear teeth crack, the mesh stiffness of the gear tooth will be decreased, thus reducing the effective bearing capacity and dynamic performance of the gear teeth. The study found that when fatigue cracks occur at the tooth roots, the cross-sectional area of the single-tooth cantilever beam changes at the cracks, which affects bending deformation, compression deformation and shear deformation, resulting in changes in the amount of deformation during gear meshing, and time-varying meshing stiffness will also change. Finally, the dynamic response also changes. Therefore, it is essential to study the calculation method of tooth meshing stiffness with crack.