## 1 Introduction  
The performance of drive systems has a substantial impact on rotorcrafts. New transmission configurations are being proposed and developed to meet the high power-to-weight ratio and high reliability requirements of helicopter main gearbox transmission systems. [PERSON] [1] proposed an alternative design solution for planetary transmissions called a split-torque design. [PERSON] [2] and [PERSON] [3, 4] concluded that the split-torque design can reduce the weight and has notable advantages over conventional designs. However, the problem remains as to how to ensure the loads are equal across the branches. Therefore, a series of projects funded by the National Aeronautics and Space Administration (NASA) [2, 5, 6, 7, 8, 9], which started in the 1980s, has conducted considerable research on cylindrical gear power-split drives and power-split transmissions containing face gears. Moreover, the cylindrical split-torque transmission and the face gear split-torque transmission were successfully applied to the Comanche [10] and Block III [11] helicopter drive systems. A critical piece of technology regarding split-torque transmissions is their equal-load design, which indicates that the loads borne in the split-torque path are as equal as possible [12]. If loads are not equal, the gears carrying larger loads are prone to excessive wear and even failure. Therefore, torque equality between different paths is a crucial topic in the design of split-torque transmissions. Scholars worldwide have extensively studied the technical requirements of load equalization in the split-torque design of cylindrical gears and face gears. The load-sharing mechanisms, statics, and dynamics of cylindrical gear split-torque have aspects studied. [PERSON] [4] demonstrated that improved load-sharing performance can be achieved through proper structural designs, which can reduce dynamic transmission errors and noise, and analyzed the dynamics of a split-torque gear system. Then, [PERSON] et al. [13] created a dynamic model of a power-split transmission, solved the motion equations using the numerical integration method, and considered how the interaction of internal parameters affects load sharing. Additionally, [PERSON] et al. [14] obtained time-varying meshing stiffness curves through loaded tooth contact analysis and set up a quasi-static model of a power-split gear train to determine the load-sharing ratios for different installation errors. [PERSON] et al. [15] modeled the nonlinear dynamics of a power-split drive system for helicopters, evaluated the load-sharing features using dynamic analysis, and improved the load-averaging performance by modifying the phase difference based on an enhanced NSGA-II optimization algorithm. [PERSON] et al. [16] constructed a static model for a power-split star gear drive containing spur and helical gears to examine the influences of machining errors, mounting errors, and component floats on the load distribution factor. Moreover, [PERSON] et al. [17] developed a two-path split-torque gear system dynamics model to analyze the natural frequency, critical speed, dynamic relative displacement of gear pairs, and load distribution characteristics. The model considered nonlinear factors, such as tooth clearance, transmission errors, and gyroscopic effects. Furthermore, [PERSON] et al. [18] analyzed loaded and unloaded tooth meshing to obtain the time-varying meshing stiffness and then constructed a dynamics model of a dual-input two-path power-sharing transmission system considering meshing errors. Then, the effects of tooth modifications on the load distributions and dynamic load characteristics were explored. A three-dimensional analysis model of a two-power-input split-torque gear drive with 41 degrees of freedom was created by [PERSON] et al. [19; 20] using the centralized parameter method.