## 2 Modeling
### Nonlinear Dynamic Model
\(\Delta e_{16}\) is the relative displacement formed by the rotation of the lower face gear along \(x_{6}\) and \(y_{6}\), expressed as:  
\[\Delta e_{16}=-r_{f}\sin\theta_{6x}\cos\beta_{1}\sin\alpha+r_{f}\sin\theta_{6y }\sin\beta_{1}\sin\alpha \tag{4}\]  
Likewise, the relative displacement \(\delta_{17}\) of the meshing point between Input Gear 1 and the upper face gear in the meshing line direction is expressed as:  
\[\delta_{17}=\left(r_{1}\theta_{1}-r_{f}\theta_{7z}+X_{1}-X_{7}\cos\beta_{1}+Y_ {7}\sin\beta_{1}\right)\cos\alpha+\left(Y_{1}-Z_{7}\right)\sin\alpha+\Delta e_ {17}-e_{17} \tag{5}\]  
\[\Delta e_{17}=r_{f}sin\theta_{7x}\cos\beta_{1}\sin\alpha-r_{f}sin\theta_{7y} \sin\beta_{1}\sin\alpha \tag{6}\]  
Similarly, the relative displacements \(\delta_{26}\) and \(\delta_{27}\) of Input Gear 2 with the lower and upper face gears in the direction of the meshing line, respectively, are:  
\[\delta_{26}=\left(r_{2}\theta_{2}-r_{f}\theta_{6z}-X_{2}+X_{6}\cos\beta_{2}-Y_ {6}\sin\beta_{2}\right)\cos\alpha+\left(Z_{6}-Y_{2}\right)\sin\alpha+\Delta e_ {26}-e_{26} \tag{7}\]  
Figure 11:0  
Figure 11:1  
Figure 11:2  
The relative displacements \(\delta_{36}\) and \(\delta_{37}\) of Idler 1 with the lower and upper face gears in the direction of the meshing line, respectively, are:  
Figure 11:3  
Figure 11:4  
Figure 11:5  
Figure 11:6The relative displacements \(\delta_{46}\) and \(\delta_{47}\) of Idler 2 with the lower and upper face gears in the direction of the meshing line, respectively, are:  
Figure 11:7  
Figure 11:8  
Figure 11:9  
Figure 10:0  
The relative displacements \(\delta_{56}\) and \(\delta_{57}\) of the tail gear with the lower and upper face gears in the direction of the meshing line, respectively, are:  
Figure 10:1  
Figure 10:2  
Figure 10:3  
Figure 10:4  
The dynamic load of the gear pair in the direction of the meshing line is obtained from Equation (23). Figure 10:5  
where \(k(t)\) is the time-varying meshing stiffness; \(\hat{\delta}\) is the relative displacement of the meshing point; \(\hat{\delta}\) is the relative velocity; and \(c_{m}\) is the meshing damping. The expression of the nonlinear backlash function \(f(\hat{\delta})\) is shown in Equation (24), where \(b\) is half of the backlash; its function graph is shown in Figure 10:6  
\[f(\delta)=\left\{ \begin{array}{l}\delta-b,\delta>b\\ 0,-b<\delta<b\\ \delta+b,\delta<-b\end{array} \right. \tag{24}\]  
\(c_{m}\) is the meshing damping, which is expressed as:  
\[c_{m}=2\xi\sqrt{k_{av}\Big{(}1/m_{eq,p}+1/m_{eq,f}\Big{)}} \tag{25}\]  
Figure 12: Schematic diagram of backlash function. where \(\xi\) is the mesh damping ratio, generally taken as 0.03-0.17; \(k_{av}\) is the mean mesh stiffness; and \(m_{ag,i}\) is the equivalent mass of the gear. Then, according to [PERSON]'s second law, the differential equation of vibration of Input Gear 1 is:  
\[\left\{ \begin{array}{l}m_{1}\ddot{X}_{1}+c_{1x}\dot{X}_{1}+k_{1x}X_{1}=F_{16} \cos\alpha-F_{17}\cos\alpha\\ m_{1}\dot{Y}_{1}+c_{1y}\dot{Y}_{1}+k_{1y}Y_{1}=F_{16}\sin\alpha-F_{17}\sin \alpha\\ f_{1}\ddot{\theta}_{1}=T_{1}-F_{16}r_{1}\cos\alpha-F_{17}r_{1}\cos\alpha\end{array} \right. \tag{26}\]  
Similarly, the differential equation for the vibration of Input Gear 2 is:  
\[\left\{ \begin{array}{l}m_{2}\ddot{X}_{2}+c_{2x}\dot{X}_{2}+k_{2x}X_{2}=F_{26} \cos\alpha-F_{27}\cos\alpha\\ m_{2}\dot{Y}_{2}+c_{2y}\dot{Y}_{2}+k_{2y}Y_{2}=F_{26}\sin\alpha-F_{27}\sin \alpha\\ f_{2}\dot{\theta}_{2}=T_{2}-F_{26}r_{2}\cos\alpha-F_{27}r_{2}\cos\alpha\end{array} \right. \tag{27}\]  
The differential equation for the vibration of Idler 1 is:  
\[\left\{ \begin{array}{l}m_{3}\ddot{X}_{3}+c_{3x}\dot{X}_{3}+k_{3x}X_{3}=-F_{36 }\cos\alpha-F_{37}\cos\alpha\\ m_{3}\dot{Y}_{3}+c_{3y}Y_{3}+k_{3y}Y_{3}=F_{36}\sin\alpha-F_{37}\sin\alpha\\ f_{3}\ddot{\theta}_{3}=F_{36}r_{3}\cos\alpha-F_{37}r_{3}\cos\alpha\end{array} \right. \tag{28}\]  
The differential equation for the vibration of Idler 2 is:  
\[\left\{