## 3 Dynamic Model and Equations of Motion
### Equations of Motion
\begin{array}{l}m_{p}(\ddot{x}_{i}-2\omega_{c}\dot{y}_{i}-\omega_{c}{} ^{2}x_{i})=(k_{spi}^{R}\cdot\Theta_{spi}^{R}+k_{spi}^{L}\cdot\Theta_{spi}^{L}) \cdot\cos\beta_{b}\cdot\sin\alpha_{t}\\ -(k_{r1 pi}^{R}\cdot\Theta_{r1 pi}^{R}+k_{r2 pi}^{L}\cdot\Theta_{r2 pi}^{L})\cdot \cos\beta_{b}\cdot\sin\alpha_{t}+k_{p}\cdot\Theta_{xspi}\\ m_{p}(\ddot{y}_{i}+2\omega_{c}\dot{x}_{i}-\omega_{c}{}^{2}y_{i})=(k_{spi}^{R} \cdot\Theta_{spi}^{R}+k_{spi}^{L}\cdot\Theta_{spi}^{L})\cdot\cos\beta_{b}\cos \alpha_{t}\\ +(k_{r1 pi}^{R}\cdot\Theta_{r1 pi}^{R}+k_{r2 pi}^{L}\cdot\Theta_{r2 pi}^{L})\cdot \cos\beta_{b}\cdot\cos\alpha_{t}+k_{p}\cdot\Theta_{xspi}\\ m_{p}\ddot{z}_{i}=k_{spi}^{R}\cdot\sin\beta_{b}\cdot\Theta_{spi}^{R}-k_{spi}^{L} \cdot\sin\beta_{b}\cdot\Theta_{spi}^{L}-k_{r1 pi}^{R}\Theta_{r1 pi}^{R}\cdot \sin\beta_{b}\\ +k_{r2 pi}^{L}\Theta_{r2 pi}^{L}\cdot\sin\beta_{b}-k_{r2 pi}\cdot\Theta_{xspi}\\ \frac{l_{p}}{r_{p}^{2}}\ddot{u}_{i}=-(k_{spi}^{R}\Theta_{spi}^{R}+k_{spi}^{L} \Theta_{spi}^{L})\cdot\cos\beta_{b}+(k_{r1 pi}^{R}\Theta_{r1 pi}^{R}+k_{r2 pi}^{L} \Theta_{r2 pi}^{L})\cdot\cos\beta_{b}\\ \end{array} \right. \tag{15}\]  
where \(m_{g}\) and \(f_{g}\) (\(g=s,p,c,r\)) represent the mass and rotational inertia of member \(g\), respectively. \(N\) refers to the plane number. \(k_{n}\) and \(k_{nz}\) are the \(x\)- or \(y\)-direction and \(z\)-direction support stiffness of member \(n\) (\(n=s,p,c,r1,r2\)), respectively. \(k_{nt}\) (\(n=r1,r2\)) respectively means the torsional support stiffness of the right and left ring gear. \(k_{spi}^{L},k_{spi}^{R},k_{r2 pi}^{L}\) and \(k_{r1 pi}^{R}\) stand for the meshing stiffness of the \(i\)th external and internal meshing pairs on the left and right sides, respectively. The motion equations of the dynamic model exhibited in Figure 2 can be rewritten in matrix form as  
\[[\mathbf{M}]\cdot\ddot{\mathbf{U}}(\mathbf{t})+\boldsymbol{\omega}_{\mathbf{c}} \cdot[\mathbf{G}]\cdot\dot{\mathbf{U}}(\mathbf{t})+\left([\mathbf{K_{b}}]+[ \mathbf{K_{m}}(\mathbf{t})]-\boldsymbol{\omega}_{\mathbf{c}}^{2}\cdot[\mathbf{K _{\Omega}}]\right)\cdot\mathbf{U}(\mathbf{t})=\mathbf{F}(\mathbf{t})+\mathbf{T} \tag{16}\]  
where \(\mathbf{U}(\mathbf{t})\),\(\dot{\mathbf{U}}(\mathbf{t})\) and \(\dot{\mathbf{U}}(\mathbf{t})\) respectively represent the HPGT generalized displacement, velocity and acceleration vectors and \(\mathbf{U}(\mathbf{t})\)=\(\{x_{s},y_{s},z_{s},u_{s},x_{r1},y_{r1},z_{r1},u_{r1},x_{r2},y_{r2},z_{r2},u_{r2},x_{c},y_{c},z_{c},u_{c},x_{1},y_{1},z_{1},u_{1},\ldots\), \(x_{N},y_{N},z_{N},u_{N}\}^{T}\). \([\mathbf{M}]\) refers to the HPGT generalized mass matrix, \([\mathbf{K_{b}}]\) is the bearing stiffness matrix. \([\mathbf{K_{m}}(\mathbf{t})]\) indicates the time-varying meshing stiffness matrix where \([\mathbf{K_{m}}(\mathbf{t})]=\left[\mathbf{K_{m}^{L}}(\mathbf{t})\right]+ \left[\mathbf{K_{m}^{R}}(\mathbf{t})\right]\).\([\mathbf{G}]\) represents the gyroscopic matrix. \([\mathbf{K_{\Omega}}]\) is the centripetal stiffness matrix. \(\mathbf{F}(\mathbf{t})\) denotes the HPGT load vector including the internal exciting force vector which arises from the static transmission error due to elastic deflections and manufacturing errors. \(\mathbf{T}\) is the HPGT external exciting force vector and \(\mathbf{T}=\{0,0,0,T_{s}/r_{s},0,0,0,0,0,0,0,0,0,0,0,-T_{c}/r_{c},0,0,0,\ldots,0,0,0 \}^{T}\), where \(T_{g}\) (\(g=s,c\)) is the external torques applied on the component g.