## 2 Modulation Effects and Dynamics Symptom of Planet Gear Fault
### Dynamic Model of Planetary Gear
Assuming the meshing error between sun and planet gear is \(e_{\mathrm{s}nt}\), accordingly, the relative displacement can be derived as  
\[\delta_{\mathrm{s}nt}=(x_{n}-x_{\mathrm{s}})\sin\varphi_{\mathrm{s}nt}+(y_{n}- y_{\mathrm{s}})\cos\varphi_{\mathrm{s}nt}+u_{\mathrm{s}}+u_{n}+e_{\mathrm{s}n}. \tag{1}\]  
Similarly, relative displacement between ring gear and planet gear is  
\[\delta_{\mathrm{s}nt}=(x_{\mathrm{r}}-x_{n})\sin\varphi_{\mathrm{s}nt}+(y_{ \mathrm{r}}-y_{n})\cos\varphi_{\mathrm{s}nt}+u_{\mathrm{r}}-u_{n}+e_{\mathrm{s }n}. \tag{2}\]  
Relative displacements between carrier and planet gear are  
\[ \begin{array}{l}\delta_{\mathrm{c}ntx}=x_{\mathrm{c}}-x_{n}-u_{\mathrm{c}} \sin\varphi_{n}\\ \delta_{\mathrm{c}nty}=y_{\mathrm{c}}-y_{n}+u_{\mathrm{c}}\cos\varphi_{n}\\ \delta_{\mathrm{c}ntu}=-\sin\varphi_{n}(x_{\mathrm{c}}-x_{n})+\cos\varphi_{n}( y_{\mathrm{c}}-y_{n})+u_{\mathrm{c}}.\end{array}  \tag{3}\]  
Dynamic functions of carrier are  
\[ \begin{array}{l}m_{\mathrm{c}}(\ddot{x}_{\mathrm{c}}-2\omega_{\mathrm{c}} \dot{y}_{\mathrm{c}}-\omega_{\mathrm{c}}^{2}x_{\mathrm{c}})+\sum\limits_{n=1}^ {NP}(k_{\mathrm{p}}\delta_{\mathrm{c}ntx}+c_{\mathrm{p}}\dot{\delta}_{\mathrm{c }ntx})+k_{\mathrm{c}}x_{\mathrm{c}}+c_{\mathrm{c}}\dot{x}_{\mathrm{c}}=0\\ m_{\mathrm{c}}(\ddot{y}_{\mathrm{c}}+2\omega_{\mathrm{c}}\dot{x}_{\mathrm{c}}- \omega_{\mathrm{c}}^{2}y_{\mathrm{c}})+\sum\limits_{n=1}^{NP}(k_{\mathrm{p}} \delta_{\mathrm{c}nty}+c_{\mathrm{p}}\dot{\delta}_{\mathrm{c}nty})+k_{ \mathrm{c}}y_{\mathrm{c}}+c_{\mathrm{c}}\dot{y}_{\mathrm{c}}=0\\ (l_{\mathrm{c}}/r_{\mathrm{c}}^{2})\ddot{u}_{\mathrm{c}}+\sum\limits_{n=1}^{NP}(k _{\mathrm{p}}\delta_{\mathrm{c}ntu}+c_{\mathrm{p}}\dot{\delta}_{\mathrm{c}ntu})+k _{\mathrm{c}t}u_{\mathrm{c}}+c_{\mathrm{c}t}\dot{u}_{\mathrm{c}}=-T_{ \mathrm{c}}/r_{\mathrm{c}}\end{array}  \tag{4}\]  
In Figure 5, under health condition, the meshing stiffness of meshing pairs vary periodically. When the planet gear meshes with ring gear or sun gear during single-tooth meshing period, meshing stiffness is smaller. When the planet gear meshes during double-tooth meshing period, meshing stiffness is larger. Duty cycle of double-tooth meshing depends on contact ratio of meshing pair. Under health condition, meshing stiffness of ring-planet meshing in each meshing cycle is identical since gear teeth are identical, as well as sun-planet meshing stiffness. When a planet gear tooth is damaged, the meshing stiffness will be smaller since the damaged gear tooth is weaker than a normal one. As shown in Figure 5a, when planet gear tooth is damaged, meshing stiffness is smaller than health condition if the damaged planet gear tooth meshes with ring gear. On the contrast, the ring-planet meshing stiffness becomes normal when the damaged planet gear tooth meshes out. The same situation could be found in Figure 5b as well. As a result, overall dynamic responses of the planetary gearbox will be influenced by planet gear fault since meshing stiffness changes when damaged planet gear tooth engages in meshing with ring gear or sun gear. Paradigm of planet gear fault diagnosis is detecting fault features of planet gear from the overall dynamic responses (usually vibration signals) by various signal processing techniques. Theoretical understanding of characteristics of overall responses helps developing fault diagnosis method. Shown in Figure 6 is simulation signals of planetary gear under health condition and fault condition. The simulation parameters are listed in Table 1.  
During the simulation, rotating frequency of the carrier is 5 Hz, which means the period of carrier rotation is 0.2 s. From Figure 6, we can see the vibration signal amplitude varies when the