# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 2 Dynamic Model of Internal Beveloid Gear System
\begin{array}{l}m_{p}\ddot{\overline{z}}_{p}-c_{m}\left[\dot{\overline{z}}_ {p}-\dot{\overline{z}}_{g}-\left(\ddot{\overline{y}}_{p}-\ddot{\overline{y}}_ {g}+R_{p}\dot{\overline{\theta}}_{p}+R_{g}\dot{\overline{\theta}}_{g}-\dot{ \overline{e}}_{y}\right)\tan\beta_{p}\right]\sin\beta_{p}+k_{pz}\overline{f}_{ b}\left(\overline{z}_{p}\right)\\ +c_{pz}\dot{\overline{z}}_{p}-k_{m}(\overline{t})f_{z}\left[\overline{z}_{p}- \overline{z}_{g}-\left(\overline{y}_{p}-\overline{y}_{g}+R_{p}\overline{ \theta}_{p}+R_{g}\overline{\theta}_{g}-\overline{e}_{y}\right)\tan\beta_{p} \right]\sin\beta_{p}=F_{a}\end{array}  \tag{22}\]  
\[ \begin{array}{l}I_{p}\ddot{\theta}_{p}+R_{p}c_{m}\left(\ddot{\overline{y}}_ {p}-\ddot{\overline{y}}_{g}+R_{p}\dot{\overline{\theta}}_{p}+R_{g}\dot{ \overline{\theta}}_{g}-\dot{\overline{e}}_{y}\right)\cos\beta_{p}\\ +R_{p}k_{m}(\overline{t})f_{y}\left(\overline{y}_{p}-\overline{y}_{g}+R_{p} \overline{\theta}_{p}+R_{g}\overline{\theta}_{g}-\overline{e}_{y}\right)\cos \beta_{p}=\overline{T}_{p}\end{array}  \tag{23}\]  
\[ \begin{array}{l}m_{g}\ddot{\overline{y}}_{g}-c_{m}\left(\ddot{\overline{y}}_ {p}-\ddot{\overline{y}}_{g}+R_{p}\dot{\overline{\theta}}_{p}+R_{g}\dot{ \overline{\theta}}_{g}-\dot{\overline{e}}_{y}\right)\cos\beta_{p}+k_{gy} \overline{f}_{b}\left(\overline{y}_{g}\right)\\ +c_{gy}\ddot{\overline{y}}_{g}-k_{m}(\overline{t})f_{y}\left(\overline{y}_{p}- \overline{y}_{g}+R_{p}\overline{\theta}_{p}+R_{g}\overline{\theta}_{g}-\dot{ \overline{e}}_{y}\right)\cos\beta_{p}=0\end{array}  \tag{24}\]  
\[ \begin{array}{l}m_{g}\ddot{\overline{z}}_{g}+c_{m}\left[\ddot{\overline{z}}_ {p}-\dot{\overline{z}}_{g}-\left(\ddot{\overline{y}}_{p}-\ddot{\overline{y}}_ {g}+R_{p}\dot{\overline{\theta}}_{p}+R_{g}\dot{\overline{\theta}}_{g}-\dot{ \overline{e}}_{y}\right)\tan\beta_{p}\right]\sin\beta_{p}+k_{gz}\overline{f}_{ b}\left(\overline{z}_{g}\right)\\ +c_{gz}\dot{\overline{z}}_{g}+k_{m}(\overline{t})f_{z}\left[\overline{z}_{p}- \overline{z}_{g}-\left(\overline{y}_{p}-\overline{y}_{g}+R_{p}\overline{ \theta}_{p}+R_{g}\overline{\theta}_{g}-\overline{e}_{y}\right)\tan\beta_{p} \right]\sin\beta_{p}=-F_{a}\end{array}  \tag{25}\]  
\[ \begin{array}{l}I_{g}\ddot{\theta}_{g}-R_{g}c_{m}\left(\ddot{\overline{y}}_ {p}-\ddot{\overline{y}}_{g}+R_{p}\dot{\overline{\theta}}_{p}+R_{g}\dot{ \overline{\theta}}_{g}-\dot{\overline{e}}_{y}\right)\cos\beta_{p}\\ -R_{g}k_{m}(\overline{t})f_{z}\left(\overline{y}_{p}-\overline{y}_{g}+R_{p} \overline{\theta}_{p}+R_{g}\overline{\theta}_{g}-\overline{e}_{y}\right)\cos \beta_{p}=-\overline{T}_{g}\end{array}  \tag{26}\]  
where \(I_{p}\), \(I_{g}\) are moment of inertia of external and internal gears (kg\(\cdot\) m\({}^{2}\)); \(m_{p}\), \(m_{g}\) are mass of external and internal gears (kg); \(c_{p}\), \(c_{g}\) are bearing support damping of external and internal gears (N\(\cdot\) s/m); \(k_{p}\), \(k_{g}\) are bearing support stiffness of external and internal gears (N\(/\)m); \(\overline{T}_{p}\), \(\overline{T}_{g}\) are input and output torque (N\(\cdot\) m); \(\theta_{p}\), \(\overline{\theta}_{g}\) are relative torsional displacement of external and internal gears (m); \(\overline{y}_{p}\), \(\overline{y}_{g}\) are relative radial displacement of external and internal gears (m); \(\overline{z}_{p}\), \(\overline{z}_{g}\) are relative axial displacement of external and internal gears (m). Table 2 shows the calculated values of bearing support stiffness and damping. Table 3 displays additional gear transmission system parameters. \begin{table}
\begin{tabular}{c c c} \hline
**Bearing Support Stiffness** & **Bearing Support Damping** \\ \hline
**Bearing Radial Support** & **Bearing Axial Support** & **Bearing Radial Support** & **Bearing Axial Support** \\
**Stiffness \(k_{y}\) (N/m)** & **Stiffness \(k_{z}\) (N/m)** & **Damping \(c_{g}\)dsad (N\(\cdot\)s/m)** & **Damping \(c_{z}\)(N\(\cdot\)s/m)** \\ \hline \(1.364\times 10^{8}\) & \(7.173\times 10^{7}\) & 2708 & 1964 \\ \hline \end{tabular}
\end{table} 
Table 2: Calculation of bearing support stiffness and damping.