# 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
#### 3.3.6 Motion Differential Equation for the Ring Gear \(r\)  
The support force of bearing acting on the left-side ring gear \(rL\) is as follows:  
\[\left\{\begin{array}{l}F_{xdRrLrR}=c_{xdRrL}(\dot{x}_{rL}-\dot{x}_{dR})+k_{ xdRrL}(x_{rL}-x_{dR})+c_{xrLrR}(\dot{x}_{rL}-\dot{x}_{rR})+k_{xrLrR}(x_{rL}-x_{ rR})\\ F_{ydRrLrR}=c_{ydRrL}(\dot{y}_{rL}-\dot{y}_{dR})+k_{ydRrL}(y_{rL}-\dot{y}_{dR})+c_{yrLrR}( \dot{y}_{rL}-\dot{y}_{rR})+k_{yrLrR}(y_{rL}-y_{rR})\\ F_{zdRrLR}=c_{zdRrL}(\dot{x}_{rL}-\dot{z}_{dR})+k_{zdRrL}(z_{rL}-\dot{z}_{dR})+c_{ xrLrR}(\dot{z}_{rL}-\dot{z}_{rR})+k_{zrLrR}(z_{rL}-z_{rR})\\ M_{\theta zdRrLrR}=c_{\theta zdRrL}(\dot{\theta}_{zrL}-\dot{\theta}_{zdR})+k_{ \theta zdRrL}(\theta_{zrL}-\dot{\theta}_{zdR})+c_{\theta zrLrR}(\dot{\theta}_{ zrL}-\dot{\theta}_{zR})+k_{\theta zrLrR}(\theta_{zrL}-\theta_{zR})\end{array}\right., \tag{27}\]  
where \(k_{xrLrR}\), \(k_{yrLrR}\), \(k_{xrLrR}\) and \(k_{\theta zrLR}\) represent the support stiffnesses of the bearing in the respective directions, and \(c_{xrLrR}\), \(c_{yrLrR}\), \(c_{zrLrR}\) and \(c_{\theta zrLrR}\) represent the support dampings of the bearing.  
The motion equation for the left-side ring gear \(rL\) is as follows:  
\[\left\{\begin{array}{l}m_{r}(\ddot{x}_{rL}-2\omega_{h}\dot{y}_{rL}-\omega_{h}^{2} x_{rL})-\sum\limits_{i=1}^{N}F_{rpil}\sin\psi_{rpi}\cos\beta_{b}+F_{xRrlrRL}=0\\ m_{r}(\ddot{y}_{rR}+2\omega_{h}\dot{x}_{rL}-\omega_{h}^{2}y_{rL})+\sum\limits_{i= 1}^{N}F_{rpil}\cos\psi_{rpi}\cos\beta_{b}+F_{yRrlrR}=0\\ m_{r}\ddot{z}_{rL}+\sum\limits_{i=1}^{N}F_{rpil}\sin\beta_{b}+F_{2 dRrLrR}=0\\ J_{r}\ddot{\theta}_{2 rL}-\sum\limits_{i=1}^{N}F_{rpil}r_{br}\cos\beta_{b}+M_{ \theta 2 dRrLrR}=0\end{array}\right., \tag{28}\]  
The support force of bearing acting on the right-side ring gear \(rR\) is as follows:  
\[\left\{\begin{array}{l}F_{bxrR}=c_{bxrR}(\dot{x}_{rR}-\omega_{h}y_{rR})+k_{ bxrR}x_{rR}\\ F_{byrR}=c_{byrR}(\dot{y}_{rR}+\omega_{h}x_{rR})+k_{byrR}y_{rR}\\ F_{bxrR}=c_{bgrR}\dot{z}_{rR}+k_{bgrR}z_{rR}\\ M_{b\theta 2 rR}=c_{b\theta 2 rR}\dot{\theta}_{2 rR}+k_{b\theta 2 rR}\theta_{2 rR} \end{array}\right., \tag{29}\]  
where \(k_{bxrR}\), \(k_{byrR}\), \(k_{bzrR}\), and \(k_{b\theta 2 rR}\) represent the support stiffnesses of the bearing in the respective directions, and \(c_{bxrR}\), \(c_{byrR}\), \(c_{bzrR}\), and \(c_{b\theta 2 rR}\) represent the support dampings of the bearing.  
The support force of the shaft acting on the right-side ring gear, \(rR\), is as follows:  
\[\left\{\begin{array}{l}F_{xrRRL}=c_{xrRRL}(\dot{x}_{rR}-\dot{x}_{rL})+k_{xrRrL }(x_{rR}-x_{rL})\\ F_{yrRRL}=c_{yrRL}(\dot{y}_{rR}-\dot{y}_{rL})+k_{yRrl}(y_{rR}-y_{rL})\\ F_{xrRrl}=c_{xrRrl}(\dot{z}_{rR}-\dot{z}_{rL})+k_{zrRrl}(z_{rR}-z_{rL})\\ M_{\theta 2 rRL}=c_{\theta 2 rRrl}(\dot{\theta}_{2 rR}-\dot{\theta}_{2 rL})+k_{ \theta 2 rRrl}(\theta_{2 rR}-\theta_{2 rL})\end{array}\right., \tag{30}\]  
The motion equation for the right-side ring gear, \(rR\), is as follows:  
\[\left\{\begin{array}{l}m_{r}(\ddot{x}_{rR}-2\omega_{h}\dot{y}_{rR}-\omega_{ h}^{2}x_{rR})-\sum\limits_{i=1}^{N}F_{rpiR}\sin\psi_{rpi}\cos\beta_{b}+F_{ bxrR}+F_{xrRrL}=0\\ m_{r}(\ddot{y}_{rR}+2\omega_{h}\dot{x}_{rR}-\omega_{h}^{2}y_{rR})+\sum\limits_{i= 1}^{N}F_{rpiR}\cos\psi_{rpi}\cos\beta_{b}+F_{byrR}+F_{yrRrl}=0\\ m_{r}\ddot{z}_{rR}-\sum\limits_{i=1}^{N}F_{rpiR}\sin\beta_{b}+F_{bzrR}+F_{zrRrL }=0\\ J_{d}\ddot{\theta}_{2 rR}-\sum\limits_{i=1}^{N}F_{rpiR}r_{br}\cos\beta_{b}+M_{ \theta 2 rR}+M_{\theta 2 rRrl}=T_{out1}\end{array}\right., \tag{31}\]