## 1 Introduction  
Spiral bevel gears (SBGs) are widely used in aerospace, automobile, and maritime industries. With the progress of science and technology, and the increasing demands in industrial applications, the study of tooth contact performance and fault characteristics of SBGs is becoming increasingly important. There are a large number of scholars who research the mesh characteristics of SBGs. [PERSON] et al. [1] employed the hyperboloid shell method to develop finite element models of both SBGs and hypoid gears, taking into account the bending characteristics of the tooth surface. The gear compliance was addressed using the Rayleigh-Ritz method, which is based on an elastic deformation solution. [PERSON] et al. [2] used slice theory to determine the load distribution and the stress distribution of a SBG, during which the SBG was divided into several spur gear pieces. However, it is difficult to consider the coupling effect between slices. [PERSON] et al. [3] developed a semi-analytical nonlinear excitation and mesh characteristic model of a SBG. Therefore, loaded tooth Contact Analysis (LTCA) is an important tooth surface simulation technology for studying spiral bevel gear mesh characteristics. It is quick to calculate the spiral bevel gear mesh characteristics under load whilst ensuring accuracy. [PERSON]'s algorithm for geometric and kinematical analysis of point contact meshing [4] is now the most popular and widely applied method. Numerous scholars [5; 6; 7; 8; 9; 10; 11; 12] calculate SBG transmission error, contact path, time-varying mesh stiffness (TVMS), and contact stress based on LTCA or improved LTCA which can further increase the calculation speed and precision. Extensive research has been conducted both domestically and internationally to investigate the impact of assembly errors on SBGs. [PERSON] et al. [13] used the finite element method to study the effects of alignment errors on SBG meshing and determine the increase under axial error of pinion and gear may lead to undesirable edge contacts. Notably, gear axial error would cause an increase in transmission error. [PERSON] et al. [14] presented the numerical procedure to simulate the loaded behavior of a hypogal gear. Experimental validation was carried out under low load, but the accuracy under higher load is not taken into account. Researchers have performed extensive studies of SBGs whilst considering assembly errors based on tooth contact analysis (TCA) and LTCA. [PERSON] et al. [15] developed a characteristic parameters analysis for the contact shape of spiral bevel gears which included assembly error. Setting the indentation depth at 0.00635 mm based on experience for LTCA calculation may diminish its universality. [PERSON] et al. [16] derived the TVMS of a SBG with cracks under offset errors. [PERSON] et al. [17] established a nonlinear Error Tooth Contact Analysis (ETCA) equation to determine how assembly errors and initial contact points affect one another. The results were compared with ease-off topography for verification to improve the calculation speed. [PERSON] [18; 19] and [PERSON] [20; 21] investigated the effects of assembly errors on contact stress, tooth root stress, and angular displacement. It came to light that assembly errors would not only worsen the conjugation of the contact tooth surface, leading to edge contact when severe, but also reduce the maximum contact stress. There have been few scholars who have conducted experiments to verify the simulation results they did. The majority of current studies on gear faults focus on spur or helical gears, with only a limited number of researchers exploring gear faults in SBGs. Furthermore, tooth tip chipping (see Figure 1), a significant aspect of SBG faults, has received very little attention in the existing literature. At present, there are four main methods for studying gear faults: (a) Finite element method (FEM). [PERSON] et al.