## 1 Introduction
The system response was obtained using the Fourier series method, which revealed the law of the variation in the friction torque coefficient and friction stiffness, as well as the impact of harmonic frequency on the transmission error. The face gear drive has the advantages of a high reduction ratio, low error sensitivity, and simple support structure, compared with cylindrical gears and spiral level gears, and is primarily utilized in split-torque transmissions for helicopter main gearboxes [6; 21]. With the support of the Advanced Rotorcraft Transmission (ART) Program, [PERSON] studied tooth geometry designs [22; 23], tooth contact analysis methods [23], modifications [24], and grinding [25] for face gear drives. The suitability of face gears for high rotational speed and heavy load transmissions was fully verified, both theoretically and experimentally, in the middle and late stages of the ART program [26; 27]. Moreover, various split-torque configurations containing face gears [5; 28; 29] have been proposed, and their load-sharing and dynamic characteristics have been thoroughly investigated. Additionally, [PERSON] et al. [27] tested six sets of face gears with different geometries and different heat treatments to evaluate the feasibility of face gears in the main helicopter rotor gearboxes and to identify the failure mechanisms and load-carrying capability in a simulated helicopter transmission environment. [PERSON] et al. [28] designed a split-power drive system with two face gears. The input pinion with a floating structure meshed with two coaxial counterrotating face gears. Moreover, [PERSON] et al. [29] proposed a quadratic split-power transmission system containing face gears. The trends in load distribution and dynamic load coefficients with tooth clearances were studied using numerical methods. [PERSON] et al. [30] examined a face gear split-power system with manufacturing and assembly errors usingthe lumped parameter model and developed a translational-torsional coupling model. An analysis of the dynamic factors' influence on the load distribution coefficient revealed that the load on the gears might be more equally distributed by adjusting the dynamic parameters. [PERSON] et al. [31] created differential equations of a two-input face gear transmission. The impacts of the eccentricity error, support stiffness, and torsional stiffness were estimated on the contact force and sharing between gear pairs. [PERSON] et al. [32] formed dynamic differential equations of a coaxial face gear drive that considered the time-varying mesh stiffness, backlash, and other factors. Moreover, the change in the load-sharing behavior under different parameters was analyzed. [PERSON] et al. [33] investigated the load-sharing behavior of a coaxial split-torque face gear train by using a combination of lumped parameters and finite elements. The analysis showed that the matching design of the backlash and support stiffness and an even number of pinion teeth can improve the system load distribution performance. [PERSON] et al. [34] explored the natural properties of the face gear power-split train and described the effect of the torsional flexibility of the input gear shaft on the dynamic load distribution. The above scholars' ongoing research has considerably advanced our understanding of dynamic gear load-sharing properties and dynamic response. However, few have modeled the load-dependent time-varying mesh stiffness of face gears and considered it in the dynamics of the CFGSTTS. Moreover, few researchers have optimized the dynamic load-sharing features of the CFGSTTS. Therefore, in this study and based on the existing research, a more accurate dynamic load-sharing analysis model for the CFGSTTS was constructed using the centralized parameters method. The model considered the load-dependent time-varying meshing stiffness and two different meshing states, the pinion driving and the face gear driving, backlash, support stiffness, and other factors.