## 1 Introduction
Unlike fixed axis gearboxes, planetary gearboxes have compound rotary motion. The planet gears rotate on their own axis around the central sun gear. The sensor signal has amplitude modulation (AM) because the interfering positions of the sun gear meshing with planet gear and planet gear meshing with ring gear with respect to the sensor change over time, which is consistent with the findings of two studies [13; 14]. The compound motion that is caused by multi-gear-meshing results in time variant vibration propagation paths, and this generates distinctive non-stationarity in the dynamic response for a planetary gearbox, so it is difficult to monitor the condition of a planetary gearbox. Characteristic frequencies directly reflect this mechanical modulation behavior, especially the sideband around the gear meshing frequency and its harmonics. The frequency and amplitude of these features vary for different operating conditions and types of gear damage, so the meshing frequency and sideband distribution that are produced by a planetary gearbox can be used to determine its state of wear. [PERSON] and [PERSON] [15], [PERSON] [16] and [PERSON] [17] determined that the involute kinematics of planetary gearboxes produce complex and asymmetric sidebands in the vibration spectrum, which is in agreement with the results of another study [18]. Previous studies [15; 16; 17; 18] did not simulate signals considering the effect of frequency modulation (FM). [PERSON] and [PERSON] [19; 20] developed a mathematical model to determine the effect of amplitude modulation (AM) and frequency modulation (FM) due to the rotation of the planet carrier and gear manufacturing errors in a dynamic model of a planetary gearbox, and they determined the sideband characteristics around meshing frequency harmonics. This improved dynamic model better predicted the sidebands in planetary gearboxes due to distributed gear errors. [PERSON] et al. [21] studied the amplitude modulation-based formula and used simple trigonometric function manipulations to determine the major physical phenomena in a planetary gearbox. This mathematical model determines the vibrational response for a planetary gearbox and is a reliable indicator of the condition of a planetary gearbox. [PERSON] et al. [22] developed a mathematical model that considered the transmission path and the direction of the meshing force to determine the mechanisms for modulation sidebands. These studies predicted and explained sidebands more accurately, but they did not consider gear fault conditions in a planetary gearbox. [PERSON] and [PERSON] [23] constructed vibration signal models to diagnose faults in a planetary gearbox. The effect of amplitude modulation and frequency modulation due to gear damage and variations in the period of time working, and the effect of the vibration transfer path were determined using the results of previous studies [15; 16; 17; 18; 19; 20; 21; 22]. The theoretical derivation was used to manually create and detect different degrees of local gear damage and naturally developed gear damage in a planetary gearbox. These frequency domain representations specifically demonstrated the presence of local cracks in the teeth of planetary gears, which are consistent with the results of another study [24]. A spectral analysis is used to diagnose transmission faults. The spectral structure of the vibration signal of a planetary gearbox and the failure symptoms of each gear are determined by a spectral analysis. This study increases the robustness of gearbox systems in industrial applications. A planetary gearbox, an embedded sensor and a virtual instrument (as shown in Figure 1) are used to log the operating signal for mechanical parts. Considering the application limitations of installation, system integration and signal accuracy, this study constructs a verification test rig and artificially seeded faulty sun, planetary and ring gears to construct a frequency spectrum using the results of previous studies [15; 16; 17; 18; 19; 20; 21; 22; 23; 24].