# Analysis of the Dynamic Characteristics of Coaxial Counter-Rotating Planetary Transmission System
## 3 Dynamic Model of the Transmission System
### Derivation of the Motion Differential Equations
#### 3.3.3 Motion Differential Equation for the Ring Gear, \(d\)  
The support force of the bearing acting on the left-side ring gear, \(dL\), is as follows:  
\[\left\{\begin{array}{l}F_{bxdL}=c_{bxdL}\dot{x}_{dL}+k_{bxdL}x_{dL}\\ F_{bydL}=c_{bydL}\dot{y}_{dL}+k_{bydL}y_{dL}\\ F_{bzzL}=c_{bzdL}\dot{z}_{dL}+k_{bzdL}z_{dL}\\ M_{b\theta zdL}=c_{b\theta zdL}\dot{\theta}_{zdL}+k_{b\theta zdL}\theta_{zdL} \end{array}\right., \tag{15}\]  
where \(k_{bxdL}\), \(k_{ydL}\), \(k_{bzzdL}\), and \(k_{b\theta zdL}\) represent the support stiffnesses of the bearing in the respective directions, and \(c_{bxdL}\), \(c_{bydL}\), \(c_{bdL}\), and \(c_{b\theta zdL}\) represent the support dampings of the bearing.  
The support force of the shaft acting on the left internal gear, \(dL\), is as follows:  
\[\left\{\begin{array}{l}F_{xdLdR}=c_{xdLdR}(\dot{x}_{dL}-\dot{x}_{dR})+k_{xdLdR }(x_{dL}-x_{dR})\\ F_{ydLdR}=c_{ydLdR}(\dot{y}_{dL}-\dot{y}_{dR})+k_{ydLdR}(y_{dL}-y_{dR})\\ F_{zdLdR}=c_{zdLdR}(\dot{z}_{dL}-\dot{z}_{dR})+k_{zdLdR}(z_{dL}-z_{dR})\\ M_{\theta zdLdR}=c_{\theta zdLdR}(\dot{\theta}_{zdL}-\theta_{zdR})+k_{\theta zdL }(\theta_{zdL}-\theta_{zdR})\end{array}\right., \tag{16}\]where \(k_{xMLdR}\), \(k_{yMLdR}\), \(k_{xMLdR}\), and \(k_{\theta 2 dRLR}\) represent the support stiffnesses of the shaft in the respective directions, and \(c_{xMLdR}\), \(c_{yMLdR}\), \(c_{zMLdR}\), and \(c_{\theta 2 dLdR}\) represent the support dampings.  
The motion equation for the left-side ring gear, \(dL\), is as follows:  
\[\left\{\begin{array}{c}m_{d}\ddot{x}_{dL}-\sum\limits_{i=1}^{N}F_{dmiL}\sin \psi_{dmi}\cos\beta_{b}+F_{bxdL}+F_{xdLdR}=0\\ m_{d}\ddot{y}_{dL}+\sum\limits_{i=1}^{N}F_{dmiL}\cos\psi_{dmi}\cos\beta_{b}+F_{ ydL}+F_{ydLdR}=0\\ m_{d}\ddot{z}_{dL}+\sum\limits_{i=1}^{N}F_{dmiL}\sin\beta_{b}+F_{bzdL}+F_{zdLdR }=0\\ J_{d}\ddot{\theta}_{2 dL}-\sum\limits_{i=1}^{N}F_{dmiL}F_{bd}\cos\beta_{b}+M_{ bdzdL}+M_{\theta 2 dLdR}=0\\ \end{array}\right., \tag{17}\]  
The support force of the shaft acting on the right-side ring gear, \(dR\), is as follows:  
\[\left\{\begin{array}{c}F_{xMLdRRL}=c_{xMLdR}(\dot{x}_{dR}-\dot{x}_{dL})+k_{ xdLdR}(x_{dR}-x_{dL})+c_{xdRFL}(\dot{x}_{dR}-\dot{x}_{rl})+k_{xdRFL}(x_{dR}-x_{rl})\\ F_{ydLdRL}=c_{yMLdR}(\dot{y}_{dR}-\dot{y}_{dL})+k_{ydLdR}(y_{dR}-y_{dL})+c_{ yMLdR}(\dot{x}_{dR}-\dot{x}_{rl})+k_{ydRFL}(y_{dR}-y_{rl})\\ F_{zdLdRL}=c_{zdLdR}(\dot{z}_{dR}-\dot{z}_{dL})+k_{zdLdR}(z_{dR}-z_{dL})+c_{ zdRFL}(\dot{z}_{dR}-z_{L})+k_{dzdRFL}(z_{dR}-z_{rl})\\ M_{\theta 2 dLdRFL}=c_{\theta 2 dLdR}(\theta_{zdR}-\theta_{zdL})+k_{\theta 2 dLdR}(\theta_{zdR}-\theta_{zdL})+c_{ \theta 2 dRFL}(\theta_{zdR}-\theta_{zL})+k_{\theta 2 dRFL}(\theta_{zdR}-\theta_{zL})\\ \end{array}\right., \tag{18}\]  
where \(k_{xMLdRL}\), \(k_{ydRFL}\), \(k_{xMLdRL}\), and \(k_{\theta 2 dRRL}\) represent the support stiffnesses of the shaft in the respective directions, and \(c_{xdRFL}\), \(c_{ydRFL}\), \(c_{zdRFL}\), and \(c_{\theta 2 dRFL}\) represent the support dampings.  
The motion equation for the right-side ring gear, \(dR\), is as follows:  
\[\left\{\begin{array}{c}m_{d}\ddot{x}_{dR}-\sum\limits_{i=1}^{N}F_{dmiR}\sin \psi_{dmi}\cos\beta_{b}+F_{xdLdRrL}=0\\ m_{d}\ddot{y}_{dR}+\sum\limits_{i=1}^{N}F_{dmiR}\cos\psi_{dmi}\cos\beta_{b}+F_{ ydLdRrL}=0\\ m_{d}\ddot{z}_{dR}-\sum\limits_{i=1}^{N}F_{dmiR}\sin\beta_{b}+F_{zdLdRrL}=0\\ J_{d}\ddot{\theta}_{2 dR}-\sum\limits_{i=1}^{N}F_{dmiR}r_{bd}\cos\beta_{b}+M_{ \theta 2 dLdRL}=0\\ \end{array}\right., \tag{19}\]