## 2 Nonlinear Dynamic Model of the Planetary Gearbox Considering Nonlinear Mesh Stiffness
For all of the healthy gear pairs,  
\[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]0  
In a gear pair, several tooth pairs may mesh at the same time. Denote all of these mesh tooth pairs and their corresponding contact status as \((\kappa_{1i},\kappa_{2i},X_{i})\). For a mesh gear pair, a mesh tooth pair that satisfies \(X<1\) always exists. The contact position of this tooth pair can be calculated by Equation (6),  
\[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]1 \[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]2  
where the subscript \(i=1\), or \(2\) represents the \(i\)-th gear of the gear pair, \(\phi_{i}\) is the gear's instantaneous rotation angular displacement and \(\phi_{i0}\) is its initial value. For a planetary gear, \(i\) in Equation (6) could take the number of the planet gear as its value. The corresponding numbers of the two teeth can be calculated by Equation (7),  
\[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]3  
where \(\kappa_{i0}\) is the number of the teeth in meshing at start and \(z_{i}\) is the number of the total teeth. For a parallel gear pair, we can set \(\kappa_{i0}=1;i=1,2\), while phase relationships among the sun-planet gear pairs and ring-planet gear pairs should be considered in a planetary gear. \(2N\) gear pairs and \(4N\) initial tooth numbers needed to be determined, where N is the total number of the planet gears. Donate these tooth numbers as \(\kappa_{i0}^{cp},i=1,2;c=r,s;p=1,..,N\), where the subscript is the gear pair, \(p\) is the number of the planet, \(r\) represents the ring and \(s\) represents the sun gear; the subscript \(i\) is the gear, in which \(i=1\) represents the center gear, i.e., the sun gear or the ring gear, and \(i=2\) represents the planet gear. Here, we set \(\kappa_{10}^{s1}=1\), \(\kappa_{10}^{r1}=1\), \(\kappa_{10}^{r1}=1\), then the other numbers can be calculated by:  
\[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]4 \[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]5 \[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]6where \(\Delta x_{p}\), \(\Delta x_{sp}\) and \(\Delta x_{rp}\) depend on the tooth number of the gears and the layout of the planet gears. In this paper, we take,  
\[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]7 \[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]8  
Based on the mesh tooth pair \((\kappa_{1},\kappa_{2},X)\) calculated by Equations (6) and (7), all of the mesh tooth pairs and their corresponding contact position \((\kappa_{1i},\kappa_{2i},X_{i})\) can be obtained by:  
\[ \begin{array}{l}\kappa_{1i}=\kappa_{1}-n,\\ \kappa_{2i}=\kappa_{2}-n,\\ X_{i}=X+n,n=0.e-X,\end{array}  \tag{10}\]  
where \(0..e-X\) represents all the integers between \(0\) and \(e-X\). Denote the total number of the teeth pairs in meshing as \(n_{mesh}\), then the instantaneous mesh stiffness of this gear pair can be calculated by:  
\[k_{m}=\sum_{i=1}^{n_{mesh}}k(X_{i})|(\kappa_{1i},\kappa_{2i}). \tag{11}\]  
Take this nonlinear mesh stiffness model of Equation (10) into the dynamic model of Equation (1); thus, the nonlinear dynamic model for a planetary gear is established. **3. Time-Varying Mesh Stiffness Model of Planetary Gears with Tooth Breakage Fault**  
In this section, a time-varying mesh stiffness model of planetary gears with tooth breakage fault is proposed. The tooth breakage fault studied in this paper is shown as Figure 2. The shape of the breakage section is assumed to be a rectangle and parallel to the axis of the gear, which is similar to the model in [21]. Define a contact rotation angular displacement \(L\) as the angular displacement of the gear from the tooth step into the meshing process to the tooth step out when it is healthy. The size of the tooth breakage \(p\) is defined as,  
\[p=1-L_{p}/L, \tag{12}\]  
where \(L\) represents the contact rotation angular displacement of a healthy tooth and \(L_{p}\) represents the loss of the contact rotation angular displacement when the tooth is broken. It can be found that the breakage size \(p\) satisfies \(0\leq p\leq 1\).