## 1 Introduction  
Gearboxes are widely applied in the power transmission field [1; 2; 3]. However, as an vital component of a gearbox, gears often operate in stressful conditions, such as excessive service load, poor lubrication, or high temperature [4], and thus some faults related to gears can be easily observed. Spalling is one of the typical gear tooth faults, which may lead to abnormal dynamic performance of the gearbox due to the reduction in gear mesh stiffness [5; 6; 7]. Therefore, the mesh stiffness computation models of gears with or without a spalling fault has garnered the increased attention of researchers in the field of fault diagnosis of gear systems [8; 9; 10]. Compared to the finite element model, many researchers preferred the analytical calculation model of mesh stiffness due to its faster calculation [11; 12]. [PERSON] [13] calculated the deformation of a spur gear under load using the theory of displacement superposition. [PERSON] [14] improved the [PERSON]'s model by considering the influence of fillet foundation on the gear. [PERSON] and [PERSON] [15] calculated the meshing stiffness of a healthy gear using the digital method. [PERSON] and [PERSON] [16] then developed the potential energy method, which was diffusely applied in the computation of gear meshing stiffness. [PERSON] [17] further optimized the model in [16], considering shear stiffness. Based on the gear tooth fillet foundation deflection correction formula put forward by [PERSON] [18], [PERSON] and [PERSON] [19] improved the original model by considering the deformation of tooth fillet foundation in the calculation model of the gear's comprehensive mesh stiffness. Aiming to study the dynamical performance of the gear system with spalling failures, some spalling fault modelling methods have been proposed [20; 21]. For instance, [PERSON] al. [22] proposed an analytical model for computing the meshing stiffness of a spalled gear, and then discussed the influences of spalling parameters on the meshing stiffness. [PERSON] et al. [23; 24] presented a dynamical model of a spalled gear pair and investigated the effect of spalling failure on the mesh stiffness and dynamic performance of a gear system. Based on the slicing method, [PERSON] [25] studied the influence of a spalling defect on meshing stiffness, and [PERSON] et al. [26] put forward a meshing stiffness calculation method for a gear with a spalled fault. [PERSON] et al. [27] investigated the vibration characteristics of a gear system with a spalled defect by establishing the dynamical model of the gear system, and then verified the accuracy of the simulation results through experiments. [PERSON] et al. [28] presented an improved mesh stiffness computation method for gears with a spalling fault by considering the influence of extended tooth contact. [PERSON] et al. [29] proposed a meshing stiffness calculation model of a spalling gear, and studied the influence of several spalled shapes, positions, and sliding friction on mesh stiffness. [PERSON] et al. [30] investigated the vibration performance of a helical gear system with spalled failure, considering the influence of sliding friction. [PERSON] et al. [21] analyzed the influence of a nonlinear elliptical tooth surface contact pattern of a spalled gear on the dynamical response of a gear system, and validated the accuracy of the developed model through experiments. [PERSON] et al. [31] put forward a new dynamical simulation model of a spur gear and verified the effectiveness of the model through various experimental tests, and then analyzed the impact of a spalled defect on the dynamical response of a gear system. However, spalling faults are usually caused by the metal shedding of the tooth surface due to the gradual propagation of the surface crack under the action of surface stress [32; 33]. As an early fault of the spalling failure, the surface crack fault is different from the spalling fault in its influence on meshing stiffness and the dynamic characteristic of the gear system.