## 1 Introduction  
Planetary gearboxes (PCs) are widely used in aerospace, wind power generation, and other fields due to their small size, smooth transmission, large transmission ratio, and high efficiency. However, due to the unique characteristics of the transmission structure, the complexity of the vibration transmission path of PCs, and the harsh working environment, its gears, bearings, and other key components are prone to different types and degrees of failure, resulting in the degradation of gearbox performance and even causing major accidents. Therefore, the fault diagnosis of PCs is of great significance in avoiding safety accidents. In recent years, the rapid development of numerical simulation technology has provided convenient and scientific solutions for clarifying the vibration spectrum structure and carrying out research on fault diagnosis methods of PCs [1]. Two different models are used to study the vibration features of PCs: the lumped-parameter dynamic model (LPDM) and the phenomenological model. Based on LPDM, nonlinear factors in the gearbox, such as shaft misalignment [2], manufacturing errors [3], backlash [4], lubrication [5], temperature [6] and friction force [7], have been investigated in depth. However, solving the equations of motion in LPDM is extremely time-consuming and complicated due to a large number of parameters and freedom degrees in this model. To reduce the parameters of the dynamics modeling process and computational complexity, scholars proposed to describe the vibration signal of the gearbox as a function related to the meshing frequency and then construct a simple and efficient phenomenological model of the gearbox [8]. [PERSON] et al. [9] proposed a representative phenomenological model to formulate and illustrate the relationship between fault-induced modulation patterns and characteristic spectral distributions. [PERSON] et al. [10] established a phenomenological model considering the time-varying speed conditions of the planetary gearbox, which effectively explained the sideband symmetry near the resonance frequency. [PERSON] et al. [11] established a phenomenological model of the planetary gearbox considering uneven load distribution among planetary gears and revealed the mechanism of additional sidebands under nonuniform load distribution conditions. [PERSON] et al. [12] found the phenomenological model and the centralized parameter model for PGs without and with faults, respectively, and compared the results of these two models to confirm that the phenomenological model is the most efficient in generating gearbox vibration signals. However, the phenomenological model has some drawbacks. Because the phenomenological model is constructed based on trigonometric functions, there is no impact component in the synthesized vibration signal, which cannot reflect the meshing impact characteristics better. To overcome this drawback and further improve the performance and adaptability of the phenomenological model, [PERSON] et al. [13;14] unified the reference points of the stiffness model and the phase model and proposed an improved phenomenological model based on the consideration of transient shock effects to realize the simultaneous characterization of the modulation characteristics and shock characteristics of the gearbox vibration signal and applied it to characterize the gearbox vibration mechanism when different sizes of faults occur on the sun gear. [PERSON] et al. [15] calculated the impact force when the gears were meshed, established the phenomenological model considering the meshing impact, compared the effects of angular displacement and meshing influence on the vibration signal of the planetary gearbox, and confirmed that the angular displacement has a more significant effect on the gearbox spectrum. In addition, [PERSON] et al. [8] improved the traditional phenomenological model by considering the gear mesh amplitude's random magnitude and the transmission path's attenuation effect, revealing the spectrum's asymmetry. [PERSON] et al.