## 2 Nonlinear Dynamic Model of the Planetary Gearbox Considering Nonlinear Mesh Stiffness
\(p=0\) means the tooth is healthy; \(0<p<1\) means half of the tooth is broken; and \(p=1\) means the tooth is totally broken. Figure 2: Sketch map for the tooth breakage fault, (**a**) variation of the tooth profile; (**b**) variation of the contact line. The mesh stiffness between the broken tooth and a healthy tooth will drop to zero at the contact position from the broken position to the top of the tooth. When the tooth breakage fault is located at the active gear, the tooth pair mesh stiffness can be modeled as:  
\[k_{p}(X)=\left\{ \begin{array}{ll}k(X),&X\leq(1-p)e\\ 0,&X>(1-p)e\end{array} \right. \tag{13}\]  
where \(k(X)\) is the healthy gear pair mesh stiffness defined in Equation (3). Taking this model into Equation (10), the nonlinear mesh stiffness model for a gear pair with one tooth broken is obtained as Equation (14),  
\[k_{m}=\sum_{i=1}^{n_{\text{mel}}}k(X_{i})|(\kappa_{1i},\kappa_{2i}),\] \[k(X_{i})|(\kappa_{1i},\kappa_{2i})=\left\{ \begin{array}{ll}k(X ),&\kappa_{2i}\
eq\kappa_{p}\\ k_{p}(X),&\kappa_{2i}=\kappa_{p}\end{array} \right., \tag{14}\]  
where \(\kappa_{p}\) is the number of the broken teeth on the planet gear. Thus, the nonlinear dynamic model for a planetary gear with one tooth broken is established. In the numerical research of the tooth breakage fault, the planetary gearbox in [29] was used. The transmission ratio of the planetary gear ratio is about 4.56, and its detailed parameters are listed in Table 1. The fault is located at the first tooth of its first planet. The natural frequencies of the gearbox calculated by the improved model are listed in Table 2. There are six rotational frequencies and six translational frequencies in total [30]. Natural frequencies will be used in the vibration signal analysis. Figure 3 shows four time-varying mesh stiffnesses in the planetary gear simulated by the stiffness model of Equation (14). The corresponding carrier rotational speed is constant in time \(\dot{\theta}_{cZ}=77\) rpm, \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Carrier** & **Ring** & **Sun** & **Planet** \\ \hline Teeth number & - & 96 & 27 & 35 \\ Mass (kg) & 5.43 & 2.35 & 0.4 & 0.66 \\ Inertia (\(10^{-3}\) kg\(\cdot\)m\({}^{2}\)) & 59.1 & 56.7 & 0.58 & 1.53 \\ Base diameter (mm) & 96.9 & 137.5 & 38.7 & 50.2 \\ Mean mesh stiffness (N/m) & & \(k_{sn}=k_{rn}=5\times 10^{8}\) \\ Bearing stiffness (N/m) & & \(k_{p}=k_{r}=k_{s}=1\times 10^{8}\) \\ Torsional stiffness (Nm/rad) & \(k_{r}w=1\times 10^{10}\); \(k_{pw}=k_{sw}=k_{cw}=0\) \\ Pressure angle (\({}^{\circ}\)) & & 24.6 \\ Helical angle (\({}^{\circ}\)) & & 0 \\ Contact ratio & & 1.24 \\ Planet number & & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the planetary gear for simulation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Rotational (Hz)** & **Translational (Hz)** \\ \hline
1 & 0 & 743.2 \\
2 & 1334.0 & 1102.7 \\
3 & 1891.3 & 1896.5 \\
4 & 2662.8 & 2276.5 \\
5 & 8045.4 & 6989.0 \\
6 & 11,822.3 & 9651.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of the planetary gear. and the breakage size \(p=1\). It can be found that periodic impulses exist in the time-varying mesh stiffness between the fault planet gear and the center gears, and a phase shift exists between the mesh stiffness impulse of the sun-planet gear pair and the impulse of the ring-planet gear pair. Figure 4 shows the mesh stiffness of the sun-fault planet gear pair with different breakage sizes. All of the carrier rotational speeds are the same, \(\dot{\theta}_{cZ}=77\) rpm. It can be found that the width of the impulse becomes bigger with the breakage size increasing. Figure 5 gives the partial enlarged views of the mesh stiffness in Figure 4, where \(A\), \(B\), \(C\), \(D\) represent different impact modes caused by the sudden change of the total number of teeth pairs in  
Figure 4: The mesh stiffness of the sun-fault planet gear pair with different breakage sizes: (**a**) \(p=0\), (**b**) \(p=0.2\), (**c**) \(p=0.5\) and (**d**) \(p=1\).