## 3 Vibration Signal Model for a Planetary Gearbox  
To address the problems of traditional sideband-related indicators, this study uses a typical planetary gear train. An illustrative configuration is shown in Figure 10. For a planetary gearbox, the ring gear is a fixed part, and the planet carrier and the planet gears are integrated together. In this study, \(r\), \(p\), \(s\) and \(H\) denote the ring gear, the planet gear, the sun gear and the planet carrier, respectively; \(N\) is the rotational speed; \(z\) is the number of gear teeth; and \(K\) is the number of planetary gears. Figure 10: Schematic drawing of the planetary gear train. Figure 9: Block diagram for data upload. The sun gear is fixed on the shaft as the high-speed input and the planet carrier is the low-speed output. The sensor is mounted on the fixed part to collect vibration data. During the operation of the gearbox, the number of meshing teeth alternates with the rotation of the gear train. When there is tooth damage in the planetary gearbox, periodic pulses or severe modulations are produced in the vibration signal model with the characteristic frequency of the damage gear and the rotational frequency of the damage gear relative to the planet carrier, as described in a previous study [24]. Vibrational signals due to damage at the meshing site are modeled as amplitude modulation and frequency modulation (AMFM) processes to determine the impact of the vibrational signal. The effects of AM and FM due to cyclical changes in operating conditions are then measured. When the damaged area meshes, the vibration signal at the meshing position is measured, excluding the effect of the transfer path on the vibration signal. The signal model is suitable for any gear system, whether it has a fixed-shaft gear or a planetary gear set. Gear meshing conditions change frequently due to errors in the manufacturing and assembly processes and gear failures. This damage creates a sudden change in the vibration signal while gear meshing. This causes amplitude modulation and frequency modulation in the original signal. Modulation is periodic and depends on the characteristic frequency of the damaged gear, so the AM and FM functions are expressed as a discrete Fourier series, for which the characteristic frequency of the damaged gear is the fundamental frequency. Therefore, the vibration signal model that is caused by the meshing of a faulty gear can be described in terms of amplitude and frequency modulation [23,24]. \[x_{0}(t)=\sum_{k=0}^{\infty}a_{k}(t)\cos[2\pi kf_{m}t+b_{k}(t)+\theta_{k}] \tag{1}\]  
\[a_{k}(t)=A_{0}[1+\sum_{n=1}^{\infty}A_{kn}\cos(2\pi nf_{c}t+\varphi_{kn})] \tag{2}\]  
\[b_{k}(t)=\sum_{l=1}^{\infty}B_{kl}\sin(2\pi lf_{c}t+\varphi_{kl}) \tag{3}\]  
where \(a_{k}(t)\) is the AM function; \(b_{k}(t)\) is the FM function; \(A_{kn}>0\) is the AM magnitudes; \(B_{kl}>0\) is the FM magnitudes; \(A_{0}\) is a dimensionless constant of the signal amplitude; \(f_{m}\) is the gear meshing frequency; \(f_{c}\) is the characteristic frequency; \(\theta\), \(\varphi\) and \(\phi\) are the initial phases of AM and FM phases, respectively; N is the highest order of AM; and L is the highest order of FM to be considered. For simplification, only the fundamental frequencies are used in (1) to (3), so only the case of \(k=n=l=1\) is considered. Therefore, (1) is simplified as:  
\[x_{0}(t)=A_{1}[1+A\cos(2\pi f_{c}t+\varphi_{1})]\cdot\cos[2\pi f_{m}t+B\sin(2 \pi f_{c}t+\varphi_{1})+\theta_{1}] \tag{4}\]  
according to the identity  
\[e^{jz\ \sin\varphi}=\sum_{k=-\infty}^{+\infty}J_{k}(z)e^{jk\varphi} \tag{5}\]  
where \(J_{k}(z)\) is a Bessel function of the first kind with integer order \(k\) and argument \(z\), so (4) can be expanded as a sum of infinite harmonics:  
\[x_{0}(t)=A_{1}[1+A\cos(2\pi f_{c}t+\varphi_{1})]\cdot\sum_{k=-\infty}^{+\infty }J_{k}(B)\cos[2\pi(f_{m}+kf_{c})t+k\varphi_{1}+\theta_{1}] \tag{6}\]Using the identities of trigonometric functions, (6) is rewritten as:  
\[