## 1 Introduction  
As early as the middle of the last century, research on face gears began [1, 2], which provided the foundation for face gear applications. Nevertheless, it has long been the case that face gears are mainly used in light load applications. It was not until the 1990s that the application of face gears in aerospace transmission mechanisms caused face gears to become a progressively hot research topic. [PERSON] [3] conducted a detailed study on the tooth surface geometry and tooth contact analysis of face gears. Subsequently, McDonnell Douglas combined a face gear split-torque drive system with a conventional planetary gear system to reduce the mass of a new helicopter transmission system by 40%, demonstrating a significant mass advantage of a face gear split-torque drive [4]. Furthermore, a sequence of tests [5, 6, 7] demonstrated that face gears can operate at high speeds and under heavy loads. [PERSON] [8] tested a face gear split-torque transmission system and showed that face gears can be effectively applied to rotorcraft gearboxes with weight-saving and cost-reduction benefits. The split-torque gear drive can realize multi-channel shunting of power and has the advantages of compact structure, small size, and being lightweight, but it also faces the problems of load sharing and fatigue life. According to the collected literature, the fatigue problem of split-torque gear transmission has been less studied. [PERSON] [9] developed a fatigue life assessment method for spur gears based on the Lundberg-Palmgren fatigue life theory and Hertz contact theory for helicopter split-torque transmission systems. As for the marine split-torque herringbone gear transmission system, [PERSON] [10] carried out a study on the correlation between vibration energy and fatigue life and obtained the variation law of fatigue life with vibration energy. On the other hand, the load-sharing problem of split-flow transmission systems has always been the focus of research. Face gears are mainly used in helicopter main reducers in the form of split-flow transmission. In the coaxial face gear split-torque transmission system, the difference in meshing stiffness caused by the difference in the upper and lower face gear structure and the existence of manufacturing error and assembly error leads to the load borne by each branch gear to be different between gears. Consequently, the research on the load-sharing characteristics has a certain reference significance for the load-sharing design of the system. The studies concerning the load sharing of split-torque drives are mainly confined to planetary gear transmission and cylindrical gear split-torque and combined power transmission. Using the contact mechanics model of planetary gears, [PERSON] [11] conducted a study on the influence of manufacturing and assembly errors of planetary carriers and gears on the load distribution amongst planets under quasi-static conditions and quantified the load-sharing behavior. [PERSON] et al. [12] modeled a 2K-H type planetary transmission system from the perspective of dynamics, calculated the floating amount of the basic components under load-splitting and equalization requirements, and analyzed the dynamic load-sharing characteristics and the effect of error variation; the analysis results showed that the floating sun wheel could improve the load sharing of the system. [PERSON] et al. [13] constructed a bending-torsion coupling dynamic model of a double-input cylindrical gear split transmission system and calculated the system load-sharing coefficient using a numerical method. The results showed that the torsional stiffness of the double gear shaft had a great influence on the load sharing. [PERSON] et al. [14] established a dynamic model of a two-stage helical planetary gear transmission system considering gear backlash and errors, studied the load-sharing performance of four load-sharing structures, and proposed a method for calculating the dynamic sensitivity of the load distribution coefficient to errors. [PERSON] et al.