## 1 Introduction  
Owing to the compact structure and high-power transmission efficiency of the transmission mechanism of the planetary gearbox, the planetary gearbox transmission mechanism has been widely used in wind turbines, automobiles, ships, aerospace, and other machinery. In terms of the planetary gear train design, the literature [1,2] proposes a new method for planetary gear train (PGT) analysis, which simplifies the design process of the planetary gear and provides guidance for practical applications. The literature [3,4] evaluated the effect of the vibration activity of gearbox housings made of composite materials and showed that housings made of composite materials have comparable or lower vibration levels at higher gear speeds and that composite gearbox housings can be a good alternative to steel solutions. The literature [5] discusses the results of a study of the original system condition monitoring of a three-stage transmission, the main objective of which was to establish the frequency characteristics of the gearing, to determine whether it is possible to perform a vibroacoustic diagnosis of the same transmission under conditions of considerable influence and to analyze the influence of the sensor mounting point on the effectiveness of the signal diagnosis. In the context of rigid body entanglement, the literature [6,7] studied the rotational motion of a body fixed at a point, with the former representing this graphically, showing the variation of the parameters of the body during the motion, and the latter using the Krylov-Bogoliubov-Mitropolski (KBM) method to obtain an approximate solution for the kinematic system, a work that is used in physics, engineering, and the aeronautical industry. With regard to the geometric model of planetary gear teeth, due to the more difficult manufacturing of the convex-concave gear profile in comparison to the involute one, an application example was defined that suppressed this disadvantage [8]. An improved algorithm for herringbone gear tooth topology modification based on the overlap rate is proposed in the literature [9], and the superiority of the improved topology-modified gear pair is verified in terms of both statics and dynamics. Planetary gearboxes are prone to failures during long-term operation, and if these failures are not detected and eliminated in time, they will cause uneven forces on the gear mesh, accelerate the wear of the gear teeth, and lead to the failure of other gears meshed with them, i.e., combined failures. Therefore, it is very necessary to establish a planetary gearbox dynamics model to study the effects of different levels of combined failures on the dynamic characteristics of planetary gear systems and to provide the necessary information for fault diagnosis and reliability studies on the planetary gearbox. In the literature [10; 11; 12; 13], gear dynamic models containing different faults were established and the mechanisms of the different faults were analyzed using the established models. The literature [14; 15; 16] derived the mesh stiffness algorithm of gears under different faults and analyzed the spectral characteristics of gear vibration signals under different fault states and transmission paths using the Fourier transform. The literature [17; 18] established a dynamic model of the gear system considering nonlinear factors such as teeth orientation errors, gear manufacturing, and assembly errors, and analyzed the dynamic response of the gear. In the literature [19; 20], the torsional vibration model with degrees of freedom and the set total parameter model were established, respectively, and the frequency characteristics of the planetary gearbox under fault-free and different fault conditions were analyzed using the proposed models. A data fusion method based on deep convolutional neural networks (DCNN) was proposed in the literature [21], and the effectiveness of the method was verified by data collected from a planetary gearbox.