## 1 Introduction  
Gear systems have been widely used in many industrial applications due to their advantages of having accurate transmission ratios, compact dimensions, and high efficiency. With increasing requirements for gear performance in cutting-edge fields, many nonlinear dynamic characteristics that affect transmission performance must be considered in gear design. Therefore, it is important to establish a relatively accurate dynamic model and investigate the complicated nonlinear behavior of the gear system subjected to various excitations. In recent years, numerous well-established dynamic models have been put forward. [PERSON] [1] analyzed the dynamic behavior of gears caused by translation-torsion coupled vibration. [PERSON] et al. [2] deduced the dynamic equations of the spur gear from an energy point of view. In addition to the torsional vibration, the pitch and yaw vibration were also considered in their model. [PERSON] [3] studied the vibration response of a multistage gear system under deterministic and random loads. [PERSON] [4] established a translation-torsion vibration model of a compound planetary gear set, focusing on the comprehensive influence of meshing stiffness, backlash, and bearing clearance on load performance. [PERSON] [5]explored the modal characteristics and dynamic response prediction and then established a bending-torsion-axis-coupled model. Meanwhile, by establishing a nine-degrees-of-freedom (nine-DOF) model that considered vibration coupling between the gearbox and the shaft, [PERSON] [6] focused on the effects of frequency, backlash, transmission error, and stiffness on nonlinear dynamic characteristics. [PERSON] [7] analyzed the influence of gear corner contact caused by the pitch deviation in gear system dynamics. [PERSON] [8] introduced the influence of geometric eccentricity and shaft-bending deformation. Although many factors have been taken into account in these works, few studies have considered the nonuniformly distributed meshing force in the dynamic modeling of gear systems. For the convenience of analysis, the meshing force is usually treated as uniformly distributed. In practical applications, however, it is nonuniformly distributed along the line of action due to machining errors, assembly errors, misalignment errors, etc. When the nonuniformly distributed meshing force is coupled with shaft deformation, the dynamic center distance, and the time-varying meshing stiffness, the transmission performance of the gear system will be seriously degraded. Therefore, the nonuniformly distributed meshing force cannot be ignored for the gear systems used in complicated working conditions, such as wind power gear systems. [PERSON] [9] analyzed the effect of misalignment error on the time-varying meshing stiffness and the transmission error and found that only the errors on the plane of action significantly influence the meshing characteristics. [PERSON] [10] studied the influence of misalignment error on the contact pattern of the gear pair and illustrated the possible harm of the nonuniformly distributed meshing force through the analysis of the contact area. Besides, [PERSON] [11] calculated the bending moment and meshing parameter of an enhanced hypoid geared rotor system under misalignment. [PERSON] [12] analyzed the influence of planetary gear position variation and bearing stiffness nonlinearity on the misalignment model of planetary gears. Furthermore, [PERSON] [13] calculated meshing force variation based on a comprehensive gear model considering various dynamic factors and analyzed system vibration under different loads and transmission errors. Nevertheless, most of them only focused on meshing force distribution and gear wear and ignored the coupling effect of the nonuniformly distributed meshing force, the shaft deformation, the dynamic center distance, and the time-varying meshing stiffness. Besides, little attention has been paid to the vibration and stability analysis of a gear system subjected to the nonuniformly distributed meshing force.