## 2 Generalized Dynamic Model
### Establishment of Global Dynamics Model
#### 2.1.3 Internal Meshing Element  
Taking the spring compression direction of the driving gear along the meshing line as positive, the vibration and torsional displacements of the three degrees of freedom of each gear are projected to the meshing line, and the dynamic model diagram of the freedom of the inner gear ring and the planetary gear in the translation and torsion direction is obtained, as shown in Figure 4.  
The vibration displacement of the internal meshing element in Figure 4 is projected to the meshing line along the degrees of freedom in each direction; that is, the overall deformation between the driven gear and the driving gear along the meshing line direction can be obtained as Equation (7).  
\[\delta_{\text{pri}}=V_{\text{i}}q_{\text{pri}}-e_{\text{pri}} \tag{7}\]  
Figure 4: Internal meshing element.  
Figure 3: External meshing element.  
where \(q_{\rm rpi}=\{x_{\rm pi},y_{\rm pi},\theta_{\rm pi},x_{\rm r},y_{\rm r},\theta_{\rm r }\}^{T}\) is the displacement column vectors of the driving gear node and the driven gear node in the internal meshing element in the global coordinate system, respectively, \(e_{\rm rpi}\) is the composite meshing error of the internal meshing gear pair along the normal direction, and \(V_{\rm i}\) is the projection vector of the transformation from the displacement of the freedom of the i-th planetary gear and ring gear in each direction to the meshing line direction, which can be expressed as Equation (8).  
\[V_{\rm i}=\left[\sin\alpha_{\rm rpi},-\cos\alpha_{\rm rpi},-r_{\rm ppi},-\sin \psi_{\rm rpi},\cos\psi_{\rm rpi},r_{\rm brief}\right] \tag{8}\]  
where \(r_{\rm bpi}\) and \(r_{\rm bpi}\) are the radius of the base circle of the planetary gear and inner ring gear, respectively, \(\psi_{\rm rpi}\) is the positive angle between the end meshing line between the i-th planetary gear and the inner ring gear and the \(Y\)-axis of the global coordinate system as Equation (9).  
\[\psi_{\rm rpi}=\alpha_{\rm rpi}\pm\varphi_{\rm pi} \tag{9}\]  
where \(\alpha_{\rm rpi}\) is the meshing angle between the i-th planetary gear and the inner ring gear, \(\varphi_{\rm pi}\) is the positive included angle between the line vector of the i-th planetary gear and the center of the inner gear ring and the X-axis of the global coordinate system. The value of \(\psi_{\rm rpi}\) is affected by the installation phase angle \(\varphi_{\rm pi}\). When the rotation direction of the driving gear is counterclockwise, the sign is positive, and vice versa.  
The motion differential equation of the internal meshing element can be obtained in the matrix form as Equation (10).  
\[M_{\rm J}\bar{q}_{\rm rpi}+C_{\rm I}(\hat{q}_{\rm rpi}-\hat{e}_{\rm rpi})+K_{ \rm I}(q_{\rm rpi}-e_{\rm rpi})=F_{\rm s} \tag{10}\]  
where \(M_{\rm J}\) is the mass matrix of the external meshing element, \(C_{\rm I}\) is the damping matrix of the external meshing element, \(K_{\rm J}\) is the stiffness matrix of the external meshing element, \(e_{\rm rpi}\), and its differential coefficient are the equivalent displacement column vector and equivalent velocity vector of the integrated meshing error decomposed in the degrees of freedom in each direction for the external meshing element, \(F_{\rm s}\) is the column vector formed by the components of the excitation force on each degree of freedom.