## 2 Generalized Dynamic Model
### Establishment of Global Dynamics Model
#### 2.1.2 External Meshing Element  
Taking the spring compression direction of the active and passive gears along the meshing line as positive, the vibration displacement and torsional displacement of the three degrees of freedom of the external meshing gear are projected to the meshing line direction, as shown in Figure 3. The relative total deformation of the projection of the gear pair along the meshing line direction \(\delta_{0}\) can be obtained as shown in Equation (3).  
\[\delta_{0}=V_{0}q_{0}-e_{0} \tag{3}\]  
where \(q_{0}=\{x_{s},y_{s},\,\theta_{s},\,x_{\text{pi}},y_{\text{pi}},\,\theta_{\text {pi}}\}\) represents the displacement column vectors of the driving gear node and the slave gear node in the external meshing element, respectively, in the global coordinate system. \(e_{0}\) represents the composite meshing error of external meshing gear pair along the normal direction. \(V_{0}\) is the projection vector of the transformation from the displacement of the degree of freedom in each direction of the external meshing gear pair to the direction of the meshing line, and its expression is shown in Equation (4).  
\[V_{o}=\left[\sin\psi_{\text{sp}},\cos\psi_{\text{sp}},r_{\text{bs}},-\sin \alpha_{\text{spi}},-\cos\alpha_{\text{spi}},-r_{\text{bp}}\right] \tag{4}\]  
where \(r_{\text{bs}}\), \(r_{\text{bp}}\) represent the base circle radius of the sun gear and the planetary gear, respectively; \(\alpha_{\text{spi}}\) represents the meshing angle between the i-th planetary gear and the sun gear; \(\psi_{\text{spi}}\) represents the angle between the meshing line between the driving gear and the i-th planetary gear and the \(Y\)-axis, the change in the driving gear's steering will affect \(\psi_{\text{spi}}\), when the driving gear's rotation is counter clockwise, the sign is positive, and vice versa. The specific expression can be expressed in Equation (5).  
\[\psi_{\text{spi}}=\alpha_{\text{spi}}\pm\varphi_{\text{pi}} \tag{5}\]  
Figure 2: Shaft element.  
where \(\psi_{\text{spi}}\) represents the engagement angle between the i-th planet gear and the driving gear, and \(\varphi_{\text{pi}}\) represents the positive included angle between the central line vector of the i-th planetary gear and the driving gear and the \(X\)-axis of the global coordinate system; that is, the installation phase angle.  
According to [PERSON]'s second law, the motion differential equation of the external meshing element can be obtained in the matrix form as Equation (6).  
\[M_{\text{o}}\ddot{q}_{\text{o}}+\text{Co}(\dot{q}_{\text{o}}-\dot{e}_{\text{o} })+K_{\text{o}}(q_{\text{o}}-e_{\text{o}})=F_{\text{o}} \tag{6}\]  
where \(M_{\text{o}}\) is the mass matrix of the external meshing element, \(C_{\text{o}}\) is the damping matrix of the external meshing element, \(K_{\text{o}}\) is the stiffness matrix of the external meshing element, \(e_{\text{o}}\) and its differential coefficient are the equivalent displacement column vector and equivalent velocity vector of the external meshing element after the integrated meshing error is decomposed in the degrees of freedom in each direction, and \(F_{\text{o}}\) is the column vector formed by the components of the excitation force on each degree of freedom.