## 3 Dynamic Model and Equations of Motion
### Equations of Motion  
Based on the dynamic models of each mesh pair as displayed in Figures 4-6, [PERSON]'s Second Law of Motion in non-inertial coordinate frames [2] is employed to acquire the equations of motion for herringbone planetary gears. The carrier's motion equations can be given by  
\[\left\{ \begin{array}{l}m_{c}(\ddot{x}_{c}-2\omega_{c}\dot{y}_{c}-{\omega_{c}}^{ 2}x_{c})=-\sum\limits_{i=1}^{N}k_{p}(\Theta_{xepi}\cdot\cos\varphi_{i}-\Theta_{ yepi}\cdot\sin\varphi_{i})-k_{c}\cdot x_{c}\\ m_{c}(\ddot{y}_{c}+2\omega_{c}\dot{x}_{c}-{\omega_{c}}^{2}y_{c})=-\sum\limits_{ i=1}^{N}k_{p}(\Theta_{xepi}\cdot\sin\varphi_{i}+\Theta_{yepi}\cdot\cos\varphi_{i})-k_{c} \cdot y_{c}\\ m_{c}\ddot{z}_{c}=\sum\limits_{i=1}^{N}k_{p2}\Theta_{xepi}-k_{c2}\cdot z_{c} \\ \frac{I_{c}}{r_{c}^{2}}\ddot{u}_{c}=-\sum\limits_{i=1}^{N}k_{p}\Theta_{yepi}-k _{ct}\cdot u_{c}-\frac{T_{c}}{r_{c}}\end{array} \right. \tag{11}\]  
In a similar manner, the equations of motion of the other components for herringbone planetary gears can be derived. The right ring gear's motion equations are represented by  
\[\left\{ \begin{array}{l}m_{r}(\ddot{x}_{r1}-2\omega_{c}\dot{y}_{r1}-{\omega_{c }}^{2}x_{r1})=\sum\limits_{i=1}^{N}k_{r1 pi}^{R}(\Theta_{r1 pi}^{R}\cdot\cos \beta_{b})\cdot\sin\varphi_{ri}-k_{r}\cdot x_{r1}\\ m_{r}(\ddot{y}_{r1}+2\omega_{c}\dot{x}_{r1}-{\omega_{c}}^{2}y_{r1})=-\sum \limits_{i=1}^{N}k_{r1 pi}^{R}(\Theta_{r1 pi}^{R}\cdot\cos\beta_{b})\cdot\cos \varphi_{ri}-k_{r}\cdot y_{r1}\\ m_{r}\ddot{z}_{r1}=\sum\limits_{i=1}^{N}k_{r1 pi}^{R}(\Theta_{r1 pi}^{R}\cdot \sin\beta_{b})-k_{r}\cdot z_{r1}\\ \frac{I_{r}}{r_{r}^{2}}\ddot{u}_{r1}=-\sum\limits_{i=1}^{N}k_{r1 pi}^{R}(\Theta _{r1 pi}^{R}\cdot\cos\beta_{b})-k_{r1t}\cdot u_{r1}\end{array} \right. \tag{12}\]  
Figure 6: Dynamic model of the \(i\)th herringbone planet–carrier pair. The left ring gear's motion equations can be expressed as  
\[\left\{ \begin{array}{l}m_{r}(\ddot{x}_{r2}-2\omega_{c}\dot{y}_{r2}-\omega_{c}{}^ {2}x_{r2})=\sum\limits_{i=1}^{N}k_{r2 pi}^{L}(\Theta_{r2 pi}^{L}\cdot\cos\beta_{b} )\sin\varphi_{ri}-k_{r}\cdot x_{r2}\\ m_{r}(\ddot{y}_{r2}+2\omega_{c}\dot{x}_{r2}-\omega_{c}{}^{2}y_{r2})=-\sum \limits_{i=1}^{N}k_{r2 pi}^{L}(\Theta_{r2 pi}^{L}\cdot\cos\beta_{b})\cos\varphi_ {ri}-k_{r}\cdot y_{r2}\\ m_{r}\ddot{z}_{r2}=-\sum\limits_{i=1}^{N}k_{r2 pi}^{L}(\Theta_{r2 pi}^{L}\cdot \sin\beta_{b})-k_{r}\cdot z_{r2}\\ \frac{l_{r}}{r^{2}}\ddot{u}_{r2}=-\sum\limits_{i=1}^{N}k_{r2 pi}^{L}(\Theta_{r2 pi}^{L}\cdot\cos\beta_{b})-k_{r2}\cdot u_{r2}\\ \end{array} \right. \tag{13}\]  
The sun gear's motion equations are given by  
\[\left\{ \begin{array}{l}m_{s}(\ddot{x}_{s}-2\omega_{c}\dot{y}_{s}-\omega_{c}{} ^{2}x_{s})=\sum\limits_{i=1}^{N}(k_{spi}^{R}\cdot\Theta_{spi}^{R}+k_{spi}^{L} \cdot\Theta_{spi}^{L})\cdot\cos\beta_{b}\cdot\sin\varphi_{si}-k_{s}\cdot x_{s} \\ m_{s}(\ddot{y}_{s}+2\omega_{c}\dot{x}_{s}-\omega_{c}{}^{2}y_{s})=-\sum \limits_{i=1}^{N}(k_{spi}^{R}\cdot\Theta_{spi}^{R}+k_{spi}^{L}\cdot\Theta_{spi}^ {L})\cdot\cos\beta_{b}\cdot\cos\varphi_{si}-k_{s}\cdot y_{s}\\ m_{s}\ddot{z}_{s}=\sum\limits_{i=1}^{N}(k_{spi}^{L}\cdot\sin\beta_{b}\cdot \Theta_{spi}^{L})-\sum\limits_{i=1}^{N}(k_{spi}^{R}\cdot\sin\beta_{b}\cdot \Theta_{spi}^{R})-k_{sz}\cdot z_{s}\\ \frac{l_{s}}{r_{s}}\ddot{u}_{s}=-\sum\limits_{i=1}^{N}(k_{spi}^{R}\cdot\Theta _{spi}^{R}\cdot\cos\beta_{b}+k_{spi}^{L}\cdot\Theta_{spi}^{L}\cdot\cos\beta_{ b})-k_{st}\cdot u_{s}+\frac{T_{s}}{r_{s}}\\ \end{array} \right. \tag{14}\]  
The \(i\)th planet-gear's motion equations can be derived as  
\[\left\{